1) The document discusses using kernel methods to test statistical dependence between random variables. It proposes a measure called constrained covariance (COCO) that finds the maximum covariance between two random variables after transforming them using reproducing kernel Hilbert spaces.
2) COCO is defined as the supremum of the covariance between smooth transformations of the random variables, where the transformations live in unit balls of RKHSs. This captures dependencies that may not be detected by simple correlation.
3) Empirically, COCO is estimated as the largest singular value of the covariance between kernel embeddings of samples from the two random variables. This provides a way to compute the proposed dependence measure in practice from data samples.
This document discusses using the maximum mean discrepancy (MMD) to compare probability distributions and samples. The MMD is defined as the distance between the mean embeddings of distributions in a reproducing kernel Hilbert space. It can be used for two sample testing to determine if samples come from the same distribution, goodness of fit testing to assess how well a model fits some data, and testing for independence. The document outlines applications of the MMD, including troubleshooting generative adversarial networks, and discusses how to construct empirical witness functions for computing the MMD from samples.
FPGA Implementation of A New Chien Search Block for Reed-Solomon Codes RS (25...IJERA Editor
The Reed-Solomon codes RS are widely used in communication systems, in particular forming part of the specification for the ETSI digital terrestrial television standard. In this paper a simple algorithm for error detection in the Chien Search block is proposed. This algorithm is based on a simple factorization of the error locator polynomial, which allows reducing the number of components required to implement the proposed algorithm on FPGA board. Consequently, it reduces the power consumption with a percentage which can reach 50 % compared to the basic RS decoder. First, we developed the design of Chien Search Block Second, we generated and simulated the hardware description language source code using Quartus software tools,finally we implemented the proposed algorithm of Chien search block for Reed-Solomon codesRS (255, 239) on FPGA board to show both the reduced hardware resources and low complexity compared to the basic algorithm.
This paper presents a novel SAT-based approach for the computation of extensions in abstract argumentation, with focus on preferred semantics, and an empirical evaluation of its performances. The approach is based on the idea of reducing the problem of computing complete extensions to a SAT problem and then using a depth-first search method to derive preferred extensions. The proposed approach has been tested using two distinct SAT solvers and compared with three state-of-the-art systems for preferred extension computation. It turns out that the proposed approach delivers significantly better performances in the large majority of the considered cases.
Analysing and combining partial problem solutions for properly informed heuri...Alexander Decker
1. The document proposes a method for generating properly informed heuristics for heuristic search algorithms like A* to solve complex problems.
2. It involves analyzing partial problem solutions, selecting some to combine, and using static backward search spaces along with dynamic forward searches on the partial problems to construct a temporary search space for heuristic generation.
3. The method was tested on Rubik's Cube problems and shown to be effective for problems in search spaces too large to explicitly enumerate, demonstrating the potential of the approach.
Algorithm Selection for Preferred Extensions EnumerationFederico Cerutti
The document discusses algorithms for enumerating preferred extensions in abstract argumentation frameworks. It compares the performance of four algorithms: AspartixM, NAD-Alg, PrefSAT, and SCC-P. It finds that algorithm selection based on graph features can accurately predict runtime, with up to 80% accuracy in classification, and improves performance over a single best solver by 2-3 times. Key discriminating features include density, number of arguments, number of strongly connected components, and features related to computing graph properties.
This document discusses pushdown automata (PDA) and provides examples. It can be summarized as follows:
(1) A PDA is like a finite automaton but with an additional stack that allows it to remember an unlimited amount of information. This makes PDAs more powerful than finite automata.
(2) The document defines PDA formally and explains their transition function and how it incorporates stack operations. Examples are provided to recognize languages like 0^n 1^n that require unlimited memory.
(3) Sample computations are shown step-by-step to demonstrate how PDAs process input strings and manipulate their stack according to the transition function. Negative examples are also included to show strings that would
This document discusses using the maximum mean discrepancy (MMD) to compare probability distributions and samples. The MMD is defined as the distance between the mean embeddings of distributions in a reproducing kernel Hilbert space. It can be used for two sample testing to determine if samples come from the same distribution, goodness of fit testing to assess how well a model fits some data, and testing for independence. The document outlines applications of the MMD, including troubleshooting generative adversarial networks, and discusses how to construct empirical witness functions for computing the MMD from samples.
FPGA Implementation of A New Chien Search Block for Reed-Solomon Codes RS (25...IJERA Editor
The Reed-Solomon codes RS are widely used in communication systems, in particular forming part of the specification for the ETSI digital terrestrial television standard. In this paper a simple algorithm for error detection in the Chien Search block is proposed. This algorithm is based on a simple factorization of the error locator polynomial, which allows reducing the number of components required to implement the proposed algorithm on FPGA board. Consequently, it reduces the power consumption with a percentage which can reach 50 % compared to the basic RS decoder. First, we developed the design of Chien Search Block Second, we generated and simulated the hardware description language source code using Quartus software tools,finally we implemented the proposed algorithm of Chien search block for Reed-Solomon codesRS (255, 239) on FPGA board to show both the reduced hardware resources and low complexity compared to the basic algorithm.
This paper presents a novel SAT-based approach for the computation of extensions in abstract argumentation, with focus on preferred semantics, and an empirical evaluation of its performances. The approach is based on the idea of reducing the problem of computing complete extensions to a SAT problem and then using a depth-first search method to derive preferred extensions. The proposed approach has been tested using two distinct SAT solvers and compared with three state-of-the-art systems for preferred extension computation. It turns out that the proposed approach delivers significantly better performances in the large majority of the considered cases.
Analysing and combining partial problem solutions for properly informed heuri...Alexander Decker
1. The document proposes a method for generating properly informed heuristics for heuristic search algorithms like A* to solve complex problems.
2. It involves analyzing partial problem solutions, selecting some to combine, and using static backward search spaces along with dynamic forward searches on the partial problems to construct a temporary search space for heuristic generation.
3. The method was tested on Rubik's Cube problems and shown to be effective for problems in search spaces too large to explicitly enumerate, demonstrating the potential of the approach.
Algorithm Selection for Preferred Extensions EnumerationFederico Cerutti
The document discusses algorithms for enumerating preferred extensions in abstract argumentation frameworks. It compares the performance of four algorithms: AspartixM, NAD-Alg, PrefSAT, and SCC-P. It finds that algorithm selection based on graph features can accurately predict runtime, with up to 80% accuracy in classification, and improves performance over a single best solver by 2-3 times. Key discriminating features include density, number of arguments, number of strongly connected components, and features related to computing graph properties.
This document discusses pushdown automata (PDA) and provides examples. It can be summarized as follows:
(1) A PDA is like a finite automaton but with an additional stack that allows it to remember an unlimited amount of information. This makes PDAs more powerful than finite automata.
(2) The document defines PDA formally and explains their transition function and how it incorporates stack operations. Examples are provided to recognize languages like 0^n 1^n that require unlimited memory.
(3) Sample computations are shown step-by-step to demonstrate how PDAs process input strings and manipulate their stack according to the transition function. Negative examples are also included to show strings that would
This document discusses pushdown automata (PDA) and provides examples. It can be summarized as:
(1) A PDA is like a finite automaton but with an additional stack that allows it to remember an infinite amount of information. Transitions can push symbols onto or pop symbols off the stack.
(2) Examples show how a PDA can recognize languages like {0^n 1^n} that require counting, which finite automata cannot do. The stack is used to match 0s and 1s.
(3) The formal definition of a PDA specifies its states, alphabet, stack alphabet, initial/accepting configurations, and transition function defining stack operations.
On Resolution Proofs for Combinational Equivalencesatrajit
The document discusses generating resolution proofs for equivalence checking of combinational circuits. It describes how modern equivalence checking engines use transformations like structural hashing, functional hashing, and rewriting to simplify circuit structures, and how these transformations can be modeled as sequences of basic operations. It proposes maintaining correspondence between circuit transformations and resolution proof fragments, such that each basic operation generates a fragment deriving new clauses.
A New Enhanced Method of Non Parametric power spectrum Estimation.CSCJournals
The spectral analysis of non uniform sampled data sequences using Fourier Periodogram method is the classical approach. In view of data fitting and computational standpoints why the Least squares periodogram(LSP) method is preferable than the “classical” Fourier periodogram and as well as to the frequently-used form of LSP due to Lomb and Scargle is explained. Then a new method of spectral analysis of nonuniform data sequences can be interpreted as an iteratively weighted LSP that makes use of a data-dependent weighting matrix built from the most recent spectral estimate. It is iterative and it makes use of an adaptive (i.e., data-dependent) weighting, we refer to it as the iterative adaptive approach (IAA).LSP and IAA are nonparametric methods that can be used for the spectral analysis of general data sequences with both continuous and discrete spectra. However, they are most suitable for data sequences with discrete spectra (i.e., sinusoidal data), which is the case we emphasize in this paper. Of the existing methods for nonuniform sinusoidal data, Welch, MUSIC and ESPRIT methods appear to be the closest in spirit to the IAA proposed here. Indeed, all these methods make use of the estimated covariance matrix that is computed in the first iteration of IAA from LSP. MUSIC and ESPRIT, on the other hand, are parametric methods that require a guess of the number of sinusoidal components present in the data, otherwise they cannot be used; furthermore.
Discrete Logarithm Problem over Prime Fields, Non-canonical Lifts and Logarit...PadmaGadiyar
This document discusses the discrete logarithm problem (DLP) over prime fields and its generalizations. It begins by defining the DLP and providing an example. It then discusses why the DLP is important as the basis for Diffie-Hellman key exchange. Various algorithms for solving the DLP are mentioned. The document goes on to discuss generalizations of the DLP to other algebraic structures like elliptic curves. It also discusses the Smart attack for solving the DLP on anomalous elliptic curves. Finally, it proposes an approach to convert the DLP modulo a prime p to the DLP modulo the composite p(p-1) by using properties of Fermat quotients and Carmichael's function.
Reducing Structural Bias in Technology Mappingsatrajit
The document discusses techniques to reduce structural bias in technology mapping. It proposes using supergates, which combine multiple library gates, to allow matches that intermediate points not present in the original circuit. It also describes performing lossless synthesis to merge equivalent networks and add choice nodes. Experimental results show the combined approach of supergates and lossless synthesis improves delay and area over the baseline.
The document summarizes informed search strategies, including best-first search algorithms like greedy search, uniform-cost search (UCS), and A* search. It provides an overview of how heuristics can be used to guide search toward more promising solutions. A* search is described as using both path cost g(n) and heuristic estimate h(n) to determine the best order of node expansion. The properties of A*, including admissibility, completeness, and optimality, are proven assuming h(n) underestimates cost to the goal. Performance depends on heuristic accuracy, with exponential growth possible if errors are large.
TMPA-2017: Generating Cost Aware Covering Arrays For Free Iosif Itkin
TMPA-2017: Tools and Methods of Program Analysis
3-4 March, 2017, Hotel Holiday Inn Moscow Vinogradovo, Moscow
Generating Cost Aware Covering Arrays For Free
Mustafa Kemal Tas, Hanefi Mercan, Gülşen Demiröz, Kamer Kaya, Cemal Yilmaz, Sabanci University
For video follow the link: https://youtu.be/Wkdd4A0rRjE
Would like to know more?
Visit our website:
www.tmpaconf.org
www.exactprosystems.com/events/tmpa
Follow us:
https://www.linkedin.com/company/exactpro-systems-llc?trk=biz-companies-cym
https://twitter.com/exactpro
Bayesian Inference and Uncertainty Quantification for Inverse ProblemsMatt Moores
So-called “inverse” problems arise when the parameters of a physical system cannot be directly observed. The mapping between these latent parameters and the space of noisy observations is represented as a mathematical model, often involving a system of differential equations. We seek to infer the parameter values that best fit our observed data. However, it is also vital to obtain accurate quantification of the uncertainty involved with these parameters, particularly when the output of the model will be used for forecasting. Bayesian inference provides well-calibrated uncertainty estimates, represented by the posterior distribution over the parameters. In this talk, I will give a brief introduction to Markov chain Monte Carlo (MCMC) algorithms for sampling from the posterior distribution and describe how they can be combined with numerical solvers for the forward model. We apply these methods to two examples of ODE models: growth curves in ecology, and thermogravimetric analysis (TGA) in chemistry. This is joint work with Matthew Berry, Mark Nelson, Brian Monaghan and Raymond Longbottom.
bayesImageS: Bayesian computation for medical Image Segmentation using a hidd...Matt Moores
This document summarizes an R package called bayesImageS that enables Bayesian computation for medical image segmentation using a hidden Potts model. It discusses the statistical model, which involves a hidden Markov random field with a Potts prior on the latent labels. Bayesian computation methods like Gibbs sampling and Metropolis-Hastings using pseudolikelihood approximation are implemented in C++ for efficiency. Experimental results demonstrate the package on a CT electron density phantom and patient radiotherapy data.
This document provides an overview of automated theorem proving. It discusses:
1) The history and background of automated theorem proving, from Hobbes and Leibniz proposing algorithmic logic to modern computer-based approaches.
2) The theoretical limitations of automated reasoning due to results like Godel's incompleteness theorems, but also practical applications like verifying mathematics and computer systems.
3) How automated reasoning involves expressing statements formally and then manipulating those expressions algorithmically, as anticipated by Leibniz centuries ago.
This document discusses approximation algorithms and introduces several combinatorial optimization problems. It begins by explaining that approximation algorithms are needed to find near-optimal solutions for problems that cannot be solved in polynomial time, such as set cover and bin packing. It then provides examples of problems that are in P, NP, and NP-complete. Several techniques for designing approximation algorithms are outlined, including greedy algorithms, linear programming, and semidefinite programming. Specific NP-complete problems like vertex cover, set cover, and independent set are introduced and approximations algorithms with performance guarantees are provided for set cover and vertex cover.
R package 'bayesImageS': a case study in Bayesian computation using Rcpp and ...Matt Moores
There are many approaches to Bayesian computation with intractable likelihoods, including the exchange algorithm, approximate Bayesian computation (ABC), thermodynamic integration, and composite likelihood. These approaches vary in accuracy as well as scalability for datasets of significant size. The Potts model is an example where such methods are required, due to its intractable normalising constant. This model is a type of Markov random field, which is commonly used for image segmentation. The dimension of its parameter space increases linearly with the number of pixels in the image, making this a challenging application for scalable Bayesian computation. My talk will introduce various algorithms in the context of the Potts model and describe their implementation in C++, using OpenMP for parallelism. I will also discuss the process of releasing this software as an open source R package on the CRAN repository.
This document provides a proof of the Kraft-McMillan theorem, which establishes a necessary condition for a code to be uniquely decodable. Specifically, it shows that for a code with codewords of lengths l1, l2, ..., lN, the inequality Σ2-li ≤ 1 must hold for the code to be uniquely decodable. The proof works by examining the nth power of the code's Kraft sum K(C) and showing that if K(C) > 1, the inequality will be violated for sufficiently large n. The document also discusses how a prefix code can be constructed given a set of codeword lengths that satisfy the Kraft-McMillan inequality.
This is concerned with designing an exact exponential time algorithm that is better than the well-known 2^n algorithm for the problem Path Contraction. This answers an open question of van't Hof et. al [TCS 2009]. This is based on the article that appeared in ICALP 2019.
The document discusses computational complexity problems that are solvable in polynomial time but for which no significantly faster algorithms are known. It presents several such problems from areas like graph algorithms, computational biology, and computational geometry. It then discusses recent work that aims to establish conditional lower bounds for the runtime of such problems by relating their hardness to standard conjectures like 3SUM, APSP, SETH, orthogonal vectors, and small universe hitting set. Fine-grained reductions are used to show relationships between problems. Overall, the document outlines an approach for proving conditional lower bounds for problems solvable in polynomial time based on reasonable complexity theoretic conjectures.
A new transformation into State Transition Algorithm for finding the global m...Michael_Chou
To promote the global search ability of the original state transition algorithm, a new operator called axesion is suggested, which aims to search along the axes and strengthen single dimensional search. Several benchmark minimization
problems are used to illustrate the advantages of the improved algorithm over other random search methods. The results of
numerical experiments show that the new transformation can enhance the performance of the state transition algorithm and the new strategy is effective and reliable.
RDFS with Attribute Equations via SPARQL RewritingStefan Bischof
RDFS with Attribute Equations via SPARQL Rewriting proposes a method to handle attribute equations in RDFS ontologies by rewriting SPARQL queries using the PerfectRef algorithm. Attribute equations are broken into adorned attributes and rewritten as UCQs. This sound and complete rewriting approach is shown to perform significantly faster on real-world datasets than using forward chaining rules in Jena, enabling reasoning over numeric properties and equations using existing SPARQL engines.
new optimization algorithm for topology optimizationSeonho Park
authors devise new convex approximation called DQA which utilizes information of two consecutive points at iterates. Also, to guarantee global convergence, filter method is illustrated.
This document discusses randomized computation and algorithms. It begins with an introduction to randomized algorithms like randomized quicksort and polynomial identity testing. It then covers computational models for probabilistic Turing machines and complexity classes like BPP, RP and ZPP that are defined using randomization. The document also characterizes these classes using quantifiers and discusses error reduction techniques for BPP.
Efficient Edge-Skeleton Computation for Polytopes Defined by OraclesVissarion Fisikopoulos
This document summarizes algorithms for computing the edge skeleton of a polytope defined by oracle functions. It first describes an existing algorithm for vertex enumeration in the oracle model that works by computing an initial simplex and recursively querying the oracle. It then presents a new algorithm for computing the edge skeleton that takes as input the oracle functions and a superset of edge directions, and works by generating candidate edge segments and validating them with the oracle. The runtime of this edge skeleton algorithm is polynomial in parameters of the polytope representation.
Convolutional networks and graph networks through kernelstuxette
This presentation discusses how convolutional kernel networks (CKNs) can be used to model sequential and graph-structured data through kernels defined over sequences and graphs. CKNs define feature maps from substructures like n-mers in sequences and paths in graphs into high-dimensional spaces, which are then approximated to obtain low-dimensional representations that can be used for prediction tasks like classification. This approach is analogous to convolutional neural networks and can be extended to multiple layers. The presentation provides examples showing CKNs achieve good performance on problems involving protein sequences and social networks.
We approach the screening problem - i.e. detecting which inputs of a computer model significantly impact the output - from a formal Bayesian model selection point of view. That is, we place a Gaussian process prior on the computer model and consider the $2^p$ models that result from assuming that each of the subsets of the $p$ inputs affect the response. The goal is to obtain the posterior probabilities of each of these models. In this talk, we focus on the specification of objective priors on the model-specific parameters and on convenient ways to compute the associated marginal likelihoods. These two problems that normally are seen as unrelated, have challenging connections since the priors proposed in the literature are specifically designed to have posterior modes in the boundary of the parameter space, hence precluding the application of approximate integration techniques based on e.g. Laplace approximations. We explore several ways of circumventing this difficulty, comparing different methodologies with synthetic examples taken from the literature.
Authors: Gonzalo Garcia-Donato (Universidad de Castilla-La Mancha) and Rui Paulo (Universidade de Lisboa)
This document discusses pushdown automata (PDA) and provides examples. It can be summarized as:
(1) A PDA is like a finite automaton but with an additional stack that allows it to remember an infinite amount of information. Transitions can push symbols onto or pop symbols off the stack.
(2) Examples show how a PDA can recognize languages like {0^n 1^n} that require counting, which finite automata cannot do. The stack is used to match 0s and 1s.
(3) The formal definition of a PDA specifies its states, alphabet, stack alphabet, initial/accepting configurations, and transition function defining stack operations.
On Resolution Proofs for Combinational Equivalencesatrajit
The document discusses generating resolution proofs for equivalence checking of combinational circuits. It describes how modern equivalence checking engines use transformations like structural hashing, functional hashing, and rewriting to simplify circuit structures, and how these transformations can be modeled as sequences of basic operations. It proposes maintaining correspondence between circuit transformations and resolution proof fragments, such that each basic operation generates a fragment deriving new clauses.
A New Enhanced Method of Non Parametric power spectrum Estimation.CSCJournals
The spectral analysis of non uniform sampled data sequences using Fourier Periodogram method is the classical approach. In view of data fitting and computational standpoints why the Least squares periodogram(LSP) method is preferable than the “classical” Fourier periodogram and as well as to the frequently-used form of LSP due to Lomb and Scargle is explained. Then a new method of spectral analysis of nonuniform data sequences can be interpreted as an iteratively weighted LSP that makes use of a data-dependent weighting matrix built from the most recent spectral estimate. It is iterative and it makes use of an adaptive (i.e., data-dependent) weighting, we refer to it as the iterative adaptive approach (IAA).LSP and IAA are nonparametric methods that can be used for the spectral analysis of general data sequences with both continuous and discrete spectra. However, they are most suitable for data sequences with discrete spectra (i.e., sinusoidal data), which is the case we emphasize in this paper. Of the existing methods for nonuniform sinusoidal data, Welch, MUSIC and ESPRIT methods appear to be the closest in spirit to the IAA proposed here. Indeed, all these methods make use of the estimated covariance matrix that is computed in the first iteration of IAA from LSP. MUSIC and ESPRIT, on the other hand, are parametric methods that require a guess of the number of sinusoidal components present in the data, otherwise they cannot be used; furthermore.
Discrete Logarithm Problem over Prime Fields, Non-canonical Lifts and Logarit...PadmaGadiyar
This document discusses the discrete logarithm problem (DLP) over prime fields and its generalizations. It begins by defining the DLP and providing an example. It then discusses why the DLP is important as the basis for Diffie-Hellman key exchange. Various algorithms for solving the DLP are mentioned. The document goes on to discuss generalizations of the DLP to other algebraic structures like elliptic curves. It also discusses the Smart attack for solving the DLP on anomalous elliptic curves. Finally, it proposes an approach to convert the DLP modulo a prime p to the DLP modulo the composite p(p-1) by using properties of Fermat quotients and Carmichael's function.
Reducing Structural Bias in Technology Mappingsatrajit
The document discusses techniques to reduce structural bias in technology mapping. It proposes using supergates, which combine multiple library gates, to allow matches that intermediate points not present in the original circuit. It also describes performing lossless synthesis to merge equivalent networks and add choice nodes. Experimental results show the combined approach of supergates and lossless synthesis improves delay and area over the baseline.
The document summarizes informed search strategies, including best-first search algorithms like greedy search, uniform-cost search (UCS), and A* search. It provides an overview of how heuristics can be used to guide search toward more promising solutions. A* search is described as using both path cost g(n) and heuristic estimate h(n) to determine the best order of node expansion. The properties of A*, including admissibility, completeness, and optimality, are proven assuming h(n) underestimates cost to the goal. Performance depends on heuristic accuracy, with exponential growth possible if errors are large.
TMPA-2017: Generating Cost Aware Covering Arrays For Free Iosif Itkin
TMPA-2017: Tools and Methods of Program Analysis
3-4 March, 2017, Hotel Holiday Inn Moscow Vinogradovo, Moscow
Generating Cost Aware Covering Arrays For Free
Mustafa Kemal Tas, Hanefi Mercan, Gülşen Demiröz, Kamer Kaya, Cemal Yilmaz, Sabanci University
For video follow the link: https://youtu.be/Wkdd4A0rRjE
Would like to know more?
Visit our website:
www.tmpaconf.org
www.exactprosystems.com/events/tmpa
Follow us:
https://www.linkedin.com/company/exactpro-systems-llc?trk=biz-companies-cym
https://twitter.com/exactpro
Bayesian Inference and Uncertainty Quantification for Inverse ProblemsMatt Moores
So-called “inverse” problems arise when the parameters of a physical system cannot be directly observed. The mapping between these latent parameters and the space of noisy observations is represented as a mathematical model, often involving a system of differential equations. We seek to infer the parameter values that best fit our observed data. However, it is also vital to obtain accurate quantification of the uncertainty involved with these parameters, particularly when the output of the model will be used for forecasting. Bayesian inference provides well-calibrated uncertainty estimates, represented by the posterior distribution over the parameters. In this talk, I will give a brief introduction to Markov chain Monte Carlo (MCMC) algorithms for sampling from the posterior distribution and describe how they can be combined with numerical solvers for the forward model. We apply these methods to two examples of ODE models: growth curves in ecology, and thermogravimetric analysis (TGA) in chemistry. This is joint work with Matthew Berry, Mark Nelson, Brian Monaghan and Raymond Longbottom.
bayesImageS: Bayesian computation for medical Image Segmentation using a hidd...Matt Moores
This document summarizes an R package called bayesImageS that enables Bayesian computation for medical image segmentation using a hidden Potts model. It discusses the statistical model, which involves a hidden Markov random field with a Potts prior on the latent labels. Bayesian computation methods like Gibbs sampling and Metropolis-Hastings using pseudolikelihood approximation are implemented in C++ for efficiency. Experimental results demonstrate the package on a CT electron density phantom and patient radiotherapy data.
This document provides an overview of automated theorem proving. It discusses:
1) The history and background of automated theorem proving, from Hobbes and Leibniz proposing algorithmic logic to modern computer-based approaches.
2) The theoretical limitations of automated reasoning due to results like Godel's incompleteness theorems, but also practical applications like verifying mathematics and computer systems.
3) How automated reasoning involves expressing statements formally and then manipulating those expressions algorithmically, as anticipated by Leibniz centuries ago.
This document discusses approximation algorithms and introduces several combinatorial optimization problems. It begins by explaining that approximation algorithms are needed to find near-optimal solutions for problems that cannot be solved in polynomial time, such as set cover and bin packing. It then provides examples of problems that are in P, NP, and NP-complete. Several techniques for designing approximation algorithms are outlined, including greedy algorithms, linear programming, and semidefinite programming. Specific NP-complete problems like vertex cover, set cover, and independent set are introduced and approximations algorithms with performance guarantees are provided for set cover and vertex cover.
R package 'bayesImageS': a case study in Bayesian computation using Rcpp and ...Matt Moores
There are many approaches to Bayesian computation with intractable likelihoods, including the exchange algorithm, approximate Bayesian computation (ABC), thermodynamic integration, and composite likelihood. These approaches vary in accuracy as well as scalability for datasets of significant size. The Potts model is an example where such methods are required, due to its intractable normalising constant. This model is a type of Markov random field, which is commonly used for image segmentation. The dimension of its parameter space increases linearly with the number of pixels in the image, making this a challenging application for scalable Bayesian computation. My talk will introduce various algorithms in the context of the Potts model and describe their implementation in C++, using OpenMP for parallelism. I will also discuss the process of releasing this software as an open source R package on the CRAN repository.
This document provides a proof of the Kraft-McMillan theorem, which establishes a necessary condition for a code to be uniquely decodable. Specifically, it shows that for a code with codewords of lengths l1, l2, ..., lN, the inequality Σ2-li ≤ 1 must hold for the code to be uniquely decodable. The proof works by examining the nth power of the code's Kraft sum K(C) and showing that if K(C) > 1, the inequality will be violated for sufficiently large n. The document also discusses how a prefix code can be constructed given a set of codeword lengths that satisfy the Kraft-McMillan inequality.
This is concerned with designing an exact exponential time algorithm that is better than the well-known 2^n algorithm for the problem Path Contraction. This answers an open question of van't Hof et. al [TCS 2009]. This is based on the article that appeared in ICALP 2019.
The document discusses computational complexity problems that are solvable in polynomial time but for which no significantly faster algorithms are known. It presents several such problems from areas like graph algorithms, computational biology, and computational geometry. It then discusses recent work that aims to establish conditional lower bounds for the runtime of such problems by relating their hardness to standard conjectures like 3SUM, APSP, SETH, orthogonal vectors, and small universe hitting set. Fine-grained reductions are used to show relationships between problems. Overall, the document outlines an approach for proving conditional lower bounds for problems solvable in polynomial time based on reasonable complexity theoretic conjectures.
A new transformation into State Transition Algorithm for finding the global m...Michael_Chou
To promote the global search ability of the original state transition algorithm, a new operator called axesion is suggested, which aims to search along the axes and strengthen single dimensional search. Several benchmark minimization
problems are used to illustrate the advantages of the improved algorithm over other random search methods. The results of
numerical experiments show that the new transformation can enhance the performance of the state transition algorithm and the new strategy is effective and reliable.
RDFS with Attribute Equations via SPARQL RewritingStefan Bischof
RDFS with Attribute Equations via SPARQL Rewriting proposes a method to handle attribute equations in RDFS ontologies by rewriting SPARQL queries using the PerfectRef algorithm. Attribute equations are broken into adorned attributes and rewritten as UCQs. This sound and complete rewriting approach is shown to perform significantly faster on real-world datasets than using forward chaining rules in Jena, enabling reasoning over numeric properties and equations using existing SPARQL engines.
new optimization algorithm for topology optimizationSeonho Park
authors devise new convex approximation called DQA which utilizes information of two consecutive points at iterates. Also, to guarantee global convergence, filter method is illustrated.
This document discusses randomized computation and algorithms. It begins with an introduction to randomized algorithms like randomized quicksort and polynomial identity testing. It then covers computational models for probabilistic Turing machines and complexity classes like BPP, RP and ZPP that are defined using randomization. The document also characterizes these classes using quantifiers and discusses error reduction techniques for BPP.
Efficient Edge-Skeleton Computation for Polytopes Defined by OraclesVissarion Fisikopoulos
This document summarizes algorithms for computing the edge skeleton of a polytope defined by oracle functions. It first describes an existing algorithm for vertex enumeration in the oracle model that works by computing an initial simplex and recursively querying the oracle. It then presents a new algorithm for computing the edge skeleton that takes as input the oracle functions and a superset of edge directions, and works by generating candidate edge segments and validating them with the oracle. The runtime of this edge skeleton algorithm is polynomial in parameters of the polytope representation.
Convolutional networks and graph networks through kernelstuxette
This presentation discusses how convolutional kernel networks (CKNs) can be used to model sequential and graph-structured data through kernels defined over sequences and graphs. CKNs define feature maps from substructures like n-mers in sequences and paths in graphs into high-dimensional spaces, which are then approximated to obtain low-dimensional representations that can be used for prediction tasks like classification. This approach is analogous to convolutional neural networks and can be extended to multiple layers. The presentation provides examples showing CKNs achieve good performance on problems involving protein sequences and social networks.
We approach the screening problem - i.e. detecting which inputs of a computer model significantly impact the output - from a formal Bayesian model selection point of view. That is, we place a Gaussian process prior on the computer model and consider the $2^p$ models that result from assuming that each of the subsets of the $p$ inputs affect the response. The goal is to obtain the posterior probabilities of each of these models. In this talk, we focus on the specification of objective priors on the model-specific parameters and on convenient ways to compute the associated marginal likelihoods. These two problems that normally are seen as unrelated, have challenging connections since the priors proposed in the literature are specifically designed to have posterior modes in the boundary of the parameter space, hence precluding the application of approximate integration techniques based on e.g. Laplace approximations. We explore several ways of circumventing this difficulty, comparing different methodologies with synthetic examples taken from the literature.
Authors: Gonzalo Garcia-Donato (Universidad de Castilla-La Mancha) and Rui Paulo (Universidade de Lisboa)
Nec 602 unit ii Random Variables and Random processDr Naim R Kidwai
The presentation explains concept of Probability, random variable, statistical averages, correlation, sum of random Variables, Central Limit Theorem,
random process, classification of random processes, power spectral density, multiple random processes.
Nonparametric testing for exogeneity with discrete regressors and instrumentsGRAPE
This document outlines a study on nonparametric testing for exogeneity with discrete regressors and instruments. It begins with an introduction and motivation for addressing endogeneity in nonparametric models. It then presents the simplest additive error model setup and discusses identification when the number of instruments is both greater than and less than the number of regressors. The document outlines two test statistics for the null hypothesis of exogeneity depending on this relationship. It also discusses estimation of the model parameters under both exogeneity and endogeneity. The summary provides a high-level overview of the key topics, models, and hypotheses discussed in the document in 3 sentences.
SYMMETRIC BILINEAR CRYPTOGRAPHY ON ELLIPTIC CURVE AND LIE ALGEBRABRNSS Publication Hub
1) The document discusses symmetric bilinear pairings on elliptic curves and Lie algebras in the context of cryptography. It provides an overview of the theoretical foundations and applications of combining these areas.
2) Key concepts covered include the Weil pairing as a symmetric bilinear pairing on elliptic curves, its properties of bilinearity and non-degeneracy, and efficient computation. Applications of elliptic curves in cryptography like ECDH and ECDSA are also summarized.
3) The security of protocols like ECDH and ECDSA relies on the assumed difficulty of solving the elliptic curve discrete logarithm problem (ECDLP). The document proves various mathematical aspects behind symmetric bilinear pairings and their use in elliptic curve cryptography.
This document summarizes the correspondence between single-layer neural networks and Gaussian processes (GPs). It reviews how the outputs of a single-layer neural network converge to a GP in the infinite-width limit, with the network's covariance function determined by its architecture. The document derives the mean and covariance functions for the GP corresponding to a single-layer network, and notes that different network outputs are independent GPs.
Iwsm2014 an analogy-based approach to estimation of software development ef...Nesma
The document discusses fuzzy analogy, a technique for software effort estimation that can handle categorical data. It introduces fuzzy analogy and fuzzy k-modes clustering. Fuzzy k-modes is used to cluster similar software projects from a repository based on categorical attributes into homogeneous groups. Fuzzy analogy then assesses the similarity between projects based on their membership to clusters and estimates the effort of a new project as a weighted average of similar past projects' efforts. The document evaluates fuzzy analogy on 194 projects from the ISBSG repository selected based on data quality and attributes criteria.
A Distributed Tableau Algorithm for Package-based Description LogicsJie Bao
The document describes a distributed tableau algorithm for reasoning with modular ontologies expressed in Package-based Description Logics (P-DL). The algorithm uses multiple local reasoners, each maintaining a local tableau for a single ontology module. Local reasoners communicate by querying each other or reporting clashes to collectively construct a global tableau without fully integrating the modules. The algorithm is proven sound and complete for P-DL with acyclic module importing. It can support reasoning across modules to answer queries.
Accelerating Metropolis Hastings with Lightweight Inference CompilationFeynman Liang
This document summarizes research on accelerating Metropolis-Hastings sampling with lightweight inference compilation. It discusses background on probabilistic programming languages and Bayesian inference techniques like variational inference and sequential importance sampling. It introduces the concept of inference compilation, where a neural network is trained to construct proposals for MCMC that better match the posterior. The paper proposes a lightweight approach to inference compilation for imperative probabilistic programs that trains proposals conditioned on execution prefixes to address issues with sequential importance sampling.
An optimal and progressive algorithm for skyline queries slideWooSung Choi
The document presents an optimal and progressive algorithm for processing skyline queries using an R-tree index. It discusses two strategies - recursive nearest neighbor queries and a branch and bound skyline algorithm. The recursive NN query approach requires additional processing to eliminate duplicate results for higher dimensions, while the branch and bound skyline algorithm prunes non-skyline points during traversal to directly generate the skyline without duplicates. The algorithm processes the R-tree in a best-first manner by maintaining a priority queue of tree nodes ordered by their minimum possible skyline size.
This document provides an overview of probabilistic reasoning and uncertainty in knowledge representation. It discusses:
1) Using probability theory to represent uncertainty quantitatively rather than logical rules with certainty factors.
2) Key concepts in probability theory including random variables, probability distributions, joint probabilities, marginal probabilities, and conditional probabilities.
3) Representing a problem domain as a probabilistic model with a sample space of possible variable states.
4) Independence of variables allowing simpler computation of probabilities.
5) The document is an introduction to probabilistic reasoning concepts to be covered in more detail later, including Bayesian networks.
This document provides an overview of probabilistic reasoning and uncertainty in knowledge representation. It discusses:
1) Using probability theory to represent uncertainty quantitatively rather than logical rules with certainty factors.
2) Key concepts in probability theory including random variables, probability distributions, joint probabilities, marginal probabilities, and conditional probabilities.
3) Representing a problem domain as a probabilistic model with a sample space of possible variable states.
4) Independence of variables allowing simpler computation of probabilities.
5) The document is an introduction to probabilistic reasoning concepts to be covered in more detail later, including Bayesian networks.
A new integer programming model for hp problemwonther
This document proposes a new integer programming model for the HP protein folding problem. [1] It formulates the HP lattice model as a graph problem to maximize the number of contacts between hydrophobic (H) amino acids. [2] The model defines binary variables to represent amino acid placement and contact edges between nodes in the lattice. [3] Computational experiments apply the model to fold two protein sequences on a 3D cubic lattice and present the results graphically.
The document discusses a robust hp-adaptation method for discontinuous Galerkin discretizations applied to aerodynamic flows. It presents a constrained pseudo-transient continuation approach to enforce physical realizability constraints during the solution process. It also describes output-based error estimation techniques to drive anisotropic hp-mesh adaptation and identify regions important for accurate output prediction. The goal is to obtain quantitatively reliable computational fluid dynamics solutions on coarse grids for engineering analysis applications.
We are interested in finding a permutation of the entries of a given square matrix so that the maximum number of its nonzero entries are moved to one of the corners in a L-shaped fashion.
If we interpret the nonzero entries of the matrix as the edges of a graph, this problem boils down to the so-called core–periphery structure, consisting of two sets: the core, a set of nodes that is highly connected across the whole graph, and the periphery, a set of nodes that is well connected only to the nodes that are in the core.
Matrix reordering problems have applications in sparse factorizations and preconditioning, while revealing core–periphery structures in networks has applications in economic, social and communication networks.
Abstract: An enhanced hybrid approach to OWL query answering that combines an RDF triple-store with an OWL reasoner in order to provide scaleable pay-as-you-go performance. The enhancements presented here include an extension to deal with arbitary OWL ontologies and optimisations that significantly improve scalability. We have implemented these techniques in a prototype system, a preliminary evaluation of which has produced very encouraging results.
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
This document provides an overview of separation logic, including:
- Applications include program analysis, verified software, and axiomatic semantics.
- Future work may focus on logics beyond pre/post conditions to specify order of actions or observable program states.
- SpaceInvader is an implementation of compositional shape analysis via bi-abduction that uses separation logic to reason about mutable data structures.
- Smallfoot is an earlier tool that used symbolic execution and a decidable fragment of separation logic to perform automatic reasoning with Hoare logic for a toy language.
This document provides an overview of information theory and coding concepts including:
1) Definitions of information, entropy, joint entropy, conditional entropy, and mutual information are introduced along with examples of calculating these quantities for discrete memoryless sources and channels.
2) Shannon's theorem for channel capacity is discussed and the channel capacity of a discrete memoryless channel is defined as the maximum mutual information over all possible input distributions.
3) Properties of entropy such as it being a measure of uncertainty, having a minimum of 0 and maximum of log2K, and being maximized when probabilities are equal are proven.
Similar to Representing and comparing probabilities: Part 2 (20)
Bayesian Non-parametric Models for Data Science using PyMCMLReview
This document provides an overview of Bayesian non-parametric models using Gaussian processes in PyMC3. It discusses the motivation for using GPs, their properties, and how they can be built and fitted in PyMC3. Examples are provided on modeling salmon recruitment data, coal mining disasters, measles outbreak data, and a multidimensional spatial dataset. Scaling issues with GPs are also addressed through sparse approximations. The PyMC3 team that develops these methods is acknowledged.
Machine Learning and Counterfactual Reasoning for "Personalized" Decision- ...MLReview
Suchi Saria
Assistant Professor
Computer Science, Applied Math & Stats and Health Policy
Institute for Computational Medicine
Hossein Soleimani
Postdoctoral Fellow Computer Science
This tutorial provides an overview of recent advances in deep generative models. It will cover three types of generative models: Markov models, latent variable models, and implicit models. The tutorial aims to give attendees a full understanding of the latest developments in generative modeling and how these models can be applied to high-dimensional data. Several challenges and open questions in the field will also be discussed. The tutorial is intended for the 2017 conference of the International Society for Bayesian Analysis.
OPTIMIZATION AS A MODEL FOR FEW-SHOT LEARNINGMLReview
This document proposes using meta-learning and an LSTM model to learn an optimization algorithm for few-shot learning. The model, called a meta-learner, is trained on multiple datasets to learn how to efficiently train a learner network on new small datasets. The meta-learner LSTM models the parameter updates of the learner network during training, learning an initialization and update rule. The inputs to the meta-learner are the loss, parameters, and gradient, and it outputs updated parameters. This learned update rule can then be used to train the learner network on new small datasets, enabling few-shot learning using only a small amount of labeled data.
2017 Tutorial - Deep Learning for Dialogue SystemsMLReview
In the past decade, goal-oriented spoken dialogue systems (SDS) have been the most promi-nent component in today’s virtual personal assistants (VPAs). Among these VPAs, Microsoft’s Cortana, Apple’s Siri, Amazon Alexa, Google Assistant, and Facebook’s M, have incorporated SDS modules in various devices, which allow users to speak naturally in order to finish tasks more efficiently. The traditional conversational systems have rather complex and/or modular pipelines. The advance of deep learning technologies has recently risen the applicatins of neural models to dialogue modeling. Nevertheless, applying deep learning technologies for building robust and scalable dialogue systems is still a challenging task and an open research area as it requires deeper understanding of the classic pipelines as well as detailed knowledge on the benchmark of the models of the prior work and the recent state-of-the-art work. Thus, this tutorial is designed to focus on an overview of the dialogue system development while describing most recent research for building dialogue systems, and summarizing the challenges. We target an audience of students and practitioners who have some deep learning background and want to get more familiar with conversational dialog systems.
Near human performance in question answering?MLReview
The document discusses a neural network system that achieves near-human performance on the SQuAD question answering dataset. However, the document argues this result requires context: SQuAD involves span selection within a paragraph where the answer is guaranteed to exist, and questions can often be answered through superficial clues without complex reasoning. While the system scores 84% F1 and humans 91%, humans were incentivized to answer quickly and their performance is underestimated. The document concludes neural networks perform clever pattern matching rather than true intelligence or reasoning, and can be easily fooled by examples beyond their training.
Tutorial on Theory and Application of Generative Adversarial NetworksMLReview
Description
Generative adversarial network (GAN) has recently emerged as a promising generative modeling approach. It consists of a generative network and a discriminative network. Through the competition between the two networks, it learns to model the data distribution. In addition to modeling the image/video distribution in computer vision problems, the framework finds use in defining visual concept using examples. To a large extent, it eliminates the need of hand-crafting objective functions for various computer vision problems. In this tutorial, we will present an overview of generative adversarial network research. We will cover several recent theoretical studies as well as training techniques and will also cover several vision applications of generative adversarial networks.
Yoav Goldberg: Word Embeddings What, How and WhitherMLReview
This document discusses word embeddings and how they work. It begins by explaining how the author became an expert in distributional semantics without realizing it. It then discusses how word2vec works, specifically skip-gram models with negative sampling. The key points are that word2vec is learning word and context vectors such that related words and contexts have similar vectors, and that this is implicitly factorizing the word-context pointwise mutual information matrix. Later sections discuss how hyperparameters are important to word2vec's success and provide critiques of common evaluation tasks like word analogies that don't capture true semantic similarity. The overall message is that word embeddings are fundamentally doing the same thing as older distributional semantic models through matrix factorization.
Physics Investigatory Project on transformers. Class 12thpihuart12
Physics investigatory project on transformers with required details for 12thes. with index, theory, types of transformers (with relevant images), procedure, sources of error, aim n apparatus along with bibliography🗃️📜. Please try to add your own imagination rather than just copy paste... Hope you all guys friends n juniors' like it. peace out✌🏻✌🏻
Mapping the Growth of Supermassive Black Holes as a Function of Galaxy Stella...Sérgio Sacani
The growth of supermassive black holes is strongly linked to their galaxies. It has been shown that the population
mean black hole accretion rate (BHAR) primarily correlates with the galaxy stellar mass (Må) and redshift for the
general galaxy population. This work aims to provide the best measurements of BHAR as a function of Må and
redshift over ranges of 109.5 < Må < 1012 Me and z < 4. We compile an unprecedentedly large sample with 8000
active galactic nuclei (AGNs) and 1.3 million normal galaxies from nine high-quality survey fields following a
wedding cake design. We further develop a semiparametric Bayesian method that can reasonably estimate BHAR
and the corresponding uncertainties, even for sparsely populated regions in the parameter space. BHAR is
constrained by X-ray surveys sampling the AGN accretion power and UV-to-infrared multiwavelength surveys
sampling the galaxy population. Our results can independently predict the X-ray luminosity function (XLF) from
the galaxy stellar mass function (SMF), and the prediction is consistent with the observed XLF. We also try adding
external constraints from the observed SMF and XLF. We further measure BHAR for star-forming and quiescent
galaxies and show that star-forming BHAR is generally larger than or at least comparable to the quiescent BHAR.
Unified Astronomy Thesaurus concepts: Supermassive black holes (1663); X-ray active galactic nuclei (2035);
Galaxies (573)
Order : Trombidiformes (Acarina) Class : Arachnida
Mites normally feed on the undersurface of the leaves but the symptoms are more easily seen on the uppersurface.
Tetranychids produce blotching (Spots) on the leaf-surface.
Tarsonemids and Eriophyids produce distortion (twist), puckering (Folds) or stunting (Short) of leaves.
Eriophyids produce distinct galls or blisters (fluid-filled sac in the outer layer)
Mechanics:- Simple and Compound PendulumPravinHudge1
a compound pendulum is a physical system with a more complex structure than a simple pendulum, incorporating its mass distribution and dimensions into its oscillatory motion around a fixed axis. Understanding its dynamics involves principles of rotational mechanics and the interplay between gravitational potential energy and kinetic energy. Compound pendulums are used in various scientific and engineering applications, such as seismology for measuring earthquakes, in clocks to maintain accurate timekeeping, and in mechanical systems to study oscillatory motion dynamics.
إتصل على هذا الرقم اذا اردت الحصول على "حبوب الاجهاض الامارات" توصيلنا مجاني رقم الواتساب 00971547952044:
00971547952044. حبوب الإجهاض في دبي | أبوظبي | الشارقة | السطوة | سعر سايتوتك Cytotec يتميز دواء Cytotec (سايتوتك) بفعاليته في إجهاض الحمل. يمكن الحصول على حبوب الاجهاض الامارات بسهولة من خلال خدمات التوصيل السريع والدفع عند الاستلام. تُستخدم حبوب سايتوتك بشكل شائع لإنهاء الحمل غير المرغوب فيه. حبوب الاجهاض الامارات هي الخيار الأمثل لمن يبحث عن طريقة آمنة وفعالة للإجهاض المنزلي.
تتوفر حبوب الاجهاض الامارات بأسعار تنافسية، ويمكنك الحصول على خصم كبير عند الشراء الآن. حبوب الاجهاض الامارات معروفة بقدرتها الفعالة على إنهاء الحمل في الشهر الأول أو الثاني. إذا كنت تبحث عن حبوب لتنزيل الحمل في الشهر الثاني أو الأول، فإن حبوب الاجهاض الامارات هي الخيار المثالي.
دواء سايتوتك يحتوي على المادة الفعالة ميزوبروستول، التي تُستخدم لإجهاض الحمل والتخلص من النزيف ما بعد الولادة. يمكنك الآن الحصول على حبوب سايتوتك للبيع في دبي وأبوظبي والشارقة من خلال الاتصال برقم 00971547952044. نسعى لتقديم أفضل الخدمات في مجال حبوب الاجهاض الامارات، مع توفير حبوب سايتوتك الأصلية بأفضل الأسعار.
إذا كنت في دبي، أبوظبي، الشارقة أو العين، يمكنك الحصول على حبوب الاجهاض الامارات بسهولة وأمان. نحن نضمن لك وصول الحبوب الأصلية بسرية تامة مع خيار الدفع عند الاستلام. حبوب الاجهاض الامارات هي الحل الفعال لإنهاء الحمل غير المرغوب فيه بطريقة آمنة.
تبحث العديد من النساء في الإمارات العربية المتحدة عن حبوب الاجهاض الامارات كبديل للعمليات الجراحية التي تتطلب وقتاً طويلاً وتكلفة عالية. بفضل حبوب الاجهاض الامارات، يمكنك الآن إنهاء الحمل بسلام وأمان في منزلك. نحن نوفر حبوب الاجهاض الامارات الأصلية من إنتاج شركة فايزر، مما يضمن لك الحصول على منتج فعال وآمن.
إذا كنت تبحث عن حبوب الاجهاض الامارات في العين، دبي، أو أبوظبي، يمكنك التواصل معنا عبر الواتس آب أو الاتصال على رقم 00971547952044 للحصول على التفاصيل حول كيفية الشراء والتوصيل. حبوب الاجهاض الامارات متوفرة بأسعار تنافسية، مع تقديم خصومات كبيرة عند الشراء بالجملة.
حبوب الاجهاض الامارات هي الخيار الأمثل لمن تبحث عن وسيلة آمنة وسريعة لإنهاء الحمل غير المرغوب فيه. تواصل معنا اليوم للحصول على حبوب الاجهاض الامارات الأصلية وتجنب أي مشاكل أو مضاعفات صحية.
في النهاية، لا تقلق بشأن الحبوب المقلدة أو الخطرة، فنحن نوفر لك حبوب الاجهاض الامارات الأصلية بأفضل الأسعار وخدمة التوصيل السريع والآمن. اتصل بنا الآن على 00971547952044 لتأكيد طلبك والحصول على حبوب الاجهاض الامارات التي تحتاجها. نحن هنا لمساعدتك وتقديم الدعم اللازم لضمان حصولك على الحل المناسب لمشكلتك.
The Limited Role of the Streaming Instability during Moon and Exomoon FormationSérgio Sacani
It is generally accepted that the Moon accreted from the disk formed by an impact between the proto-Earth and
impactor, but its details are highly debated. Some models suggest that a Mars-sized impactor formed a silicate
melt-rich (vapor-poor) disk around Earth, whereas other models suggest that a highly energetic impact produced a
silicate vapor-rich disk. Such a vapor-rich disk, however, may not be suitable for the Moon formation, because
moonlets, building blocks of the Moon, of 100 m–100 km in radius may experience strong gas drag and fall onto
Earth on a short timescale, failing to grow further. This problem may be avoided if large moonlets (?100 km)
form very quickly by streaming instability, which is a process to concentrate particles enough to cause gravitational
collapse and rapid formation of planetesimals or moonlets. Here, we investigate the effect of the streaming
instability in the Moon-forming disk for the first time and find that this instability can quickly form ∼100 km-sized
moonlets. However, these moonlets are not large enough to avoid strong drag, and they still fall onto Earth quickly.
This suggests that the vapor-rich disks may not form the large Moon, and therefore the models that produce vaporpoor disks are supported. This result is applicable to general impact-induced moon-forming disks, supporting the
previous suggestion that small planets (<1.6 R⊕) are good candidates to host large moons because their impactinduced disks would likely be vapor-poor. We find a limited role of streaming instability in satellite formation in an
impact-induced disk, whereas it plays a key role during planet formation.
Unified Astronomy Thesaurus concepts: Earth-moon system (436)
Compositions of iron-meteorite parent bodies constrainthe structure of the pr...Sérgio Sacani
Magmatic iron-meteorite parent bodies are the earliest planetesimals in the Solar System,and they preserve information about conditions and planet-forming processes in thesolar nebula. In this study, we include comprehensive elemental compositions andfractional-crystallization modeling for iron meteorites from the cores of five differenti-ated asteroids from the inner Solar System. Together with previous results of metalliccores from the outer Solar System, we conclude that asteroidal cores from the outerSolar System have smaller sizes, elevated siderophile-element abundances, and simplercrystallization processes than those from the inner Solar System. These differences arerelated to the formation locations of the parent asteroids because the solar protoplane-tary disk varied in redox conditions, elemental distributions, and dynamics at differentheliocentric distances. Using highly siderophile-element data from iron meteorites, wereconstruct the distribution of calcium-aluminum-rich inclusions (CAIs) across theprotoplanetary disk within the first million years of Solar-System history. CAIs, the firstsolids to condense in the Solar System, formed close to the Sun. They were, however,concentrated within the outer disk and depleted within the inner disk. Future modelsof the structure and evolution of the protoplanetary disk should account for this dis-tribution pattern of CAIs.
BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptxgoluk9330
Ahota Beel, nestled in Sootea Biswanath Assam , is celebrated for its extraordinary diversity of bird species. This wetland sanctuary supports a myriad of avian residents and migrants alike. Visitors can admire the elegant flights of migratory species such as the Northern Pintail and Eurasian Wigeon, alongside resident birds including the Asian Openbill and Pheasant-tailed Jacana. With its tranquil scenery and varied habitats, Ahota Beel offers a perfect haven for birdwatchers to appreciate and study the vibrant birdlife that thrives in this natural refuge.
Continuing with the partner Introduction, Tampere University has another group operating at the INSIGHT project! Meet members of the Industrial Engineering and Management Unit - Aki, Jaakko, Olga, and Vilma!
Hariyalikart Case Study of helping farmers in Biharrajsaurav589
Helping farmers all across India through our latest technologies of modern farming like drones for irrigation and best pest control For more visit : https://www.hariyalikart.com/case-study
3. Statistical model criticism
MMD@P;QA a kf £k2 a supkf kF 1‘EQ f Epf “
-4 -2 2 4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
p(x)
q(x)
f *
(x)
f £@xA is the witness function
Can we compute MMD with samples from Q and a model P?
Problem: usualy can’t compute Epf in closed form.
3/52
4. Stein idea
To get rid of Epf in
sup
kf kF 1
‘Eq f Epf “
we define the Stein operator
‘Tpf “@xA a 1
p@xA
d
dx
@f @xAp@xAA
Then
EP TP f a 0
subject to appropriate boundary conditions. (Oates, Girolami, Chopin, 2016)
4/52
5. Stein idea: proof
Ep ‘Tpf “ a
Z
1
p@xA
d
dx
@f @xAp@xAA
p@xAdx
Z
d
dx
@f @xAp@xAA
dx
a ‘f @xAp@xA“I
I
a 0
5/52
6. Stein idea: proof
Ep ‘Tpf “ a
Z
1
¨¨¨p@xA
d
dx
@f @xAp@xAA
¨
¨¨p@xAdx
Z
d
dx
@f @xAp@xAA
dx
a ‘f @xAp@xA“I
I
a 0
5/52
7. Stein idea: proof
Ep ‘Tpf “ a
Z
1
¨¨¨p@xA
d
dx
@f @xAp@xAA
¨
¨¨p@xAdx
Z
d
dx
@f @xAp@xAA
dx
a ‘f @xAp@xA“I
I
a 0
5/52
8. Stein idea: proof
Ep ‘Tpf “ a
Z
1
¨¨¨p@xA
d
dx
@f @xAp@xAA
¨
¨¨p@xAdx
Z
d
dx
@f @xAp@xAA
dx
a ‘f @xAp@xA“I
I
a 0
5/52
9. Stein idea: proof
Ep ‘Tpf “ a
Z
1
¨¨¨p@xA
d
dx
@f @xAp@xAA
¨
¨¨p@xAdx
Z
d
dx
@f @xAp@xAA
dx
a ‘f @xAp@xA“I
I
a 0
5/52
10. Kernel Stein Discrepancy
Stein operator
Tpf a @x f Cf @x @log pA
Kernel Stein Discrepancy (KSD)
KSD@p;q;pA a sup
kgkF 1
Eq Tpg EpTpg
6/52
11. Kernel Stein Discrepancy
Stein operator
Tpf a @x f Cf @x @log pA
Kernel Stein Discrepancy (KSD)
KSD@p;q;pA a sup
kgkF 1
Eq Tpg $$$$EpTpg a sup
kgkF 1
Eq Tpg
6/52
12. Kernel Stein Discrepancy
Stein operator
Tpf a @x f Cf @x @log pA
Kernel Stein Discrepancy (KSD)
KSD@p;q;pA a sup
kgkF 1
Eq Tpg $$$$EpTpg a sup
kgkF 1
Eq Tpg
-4 -2 2 4
-0.6
-0.4
-0.2
0.2
0.4
p(x)
q(x)
g*
(x)
6/52
13. Kernel Stein Discrepancy
Stein operator
Tpf a @x f Cf @x @log pA
Kernel Stein Discrepancy (KSD)
KSD@p;q;pA a sup
kgkF 1
Eq Tpg $$$$EpTpg a sup
kgkF 1
Eq Tpg
-4 -2 2 4
0.1
0.2
0.3
0.4
p(x)
q(x)
g*
(x)
6/52
14. Kernel stein discrepancy
Closed-form expression for KSD: given Z;ZH $ q, then
(Chwialkowski, Strathmann, G., ICML 2016) (Liu, Lee, Jordan ICML 2016)
KSD@p;q;pA a Eq hp@Z;ZHA
where
hp@x;yA Xa @x log p@xA@x log p@yAk@x;yA
C@y log p@yA@x k@x;yA
C@x log p@xA@y k@x;yA
C@x @y k@x;yA
and k is RKHS kernel for p
Only depends on kernel and @x log p(x). Do not need to
normalize p, or sample from it.
7/52
19. Statistical model criticism
Chicago crime data
Model is Gaussian mixture with ten components
Stein witness function
Code: https://github.com/karlnapf/kernel_goodness_of_fit 8/52
20. Kernel stein discrepancy
Further applications:
Evaluation of approximate MCMC methods.
(Chwialkowski, Strathmann, G., ICML 2016; Gorham, Mackey, ICML 2017)
What kernel to use?
The inverse multiquadric kernel,
k@x;yA a
c Ckx yk2
2
24. Dependence testing
Given: Samples from a distribution PX Y
Goal: Are X and Y independent?
Their noses guide them
through life, and they're
never happier than when
following an interesting scent.
A large animal who slings slobber,
exudes a distinctive houndy odor,
and wants nothing more than to
follow his nose.
Text from dogtime.com and petfinder.com
A responsive, interactive
pet, one that will blow in
your ear and follow you
everywhere.
YX
11/52
25. MMD as a dependence measure?
Could we use MMD?
MMD@PXY
| {z }
P
;PX PY
| {z }
Q
;rA
We don’t have samples from Q Xa PX PY , only pairs
f@xi ;yi gn
i=1
i:i:d:
$ PXY
Solution: simulate Q with pairs (xi ;yj ) for j T= i.
What kernel to use for the RKHS r?
12/52
26. MMD as a dependence measure?
Could we use MMD?
MMD@PXY
| {z }
P
;PX PY
| {z }
Q
;rA
We don’t have samples from Q Xa PX PY , only pairs
f@xi ;yi gn
i=1
i:i:d:
$ PXY
Solution: simulate Q with pairs (xi ;yj ) for j T= i.
What kernel to use for the RKHS r?
12/52
27. MMD as a dependence measure?
Could we use MMD?
MMD@PXY
| {z }
P
;PX PY
| {z }
Q
;rA
We don’t have samples from Q Xa PX PY , only pairs
f@xi ;yi gn
i=1
i:i:d:
$ PXY
Solution: simulate Q with pairs (xi ;yj ) for j T= i.
What kernel to use for the RKHS r?
12/52
28. MMD as a dependence measure
Kernel k on images with feature space p,
Kernel l on captions with feature space q,
13/52
29. MMD as a dependence measure
Kernel k on images with feature space p,
Kernel l on captions with feature space q,
Kernel on image-text pairs: are images and captions similar?
13/52
30. MMD as a dependence measure
Given: Samples from a distribution PX Y
Goal: Are X and Y independent?
MMD2
@bPXY ; bPX
bPY ;rA Xa 1
n2
trace@KLA
( K, L column centered)
14/52
31. MMD as a dependence measure
Given: Samples from a distribution PX Y
Goal: Are X and Y independent?
MMD2
@bPXY ; bPX
bPY ;rA Xa 1
n2
trace@KLA
14/52
32. MMD as a dependence measure
Two questions:
Why the product kernel? Many ways to combine kernels - why not
eg a sum?
Is there a more interpretable way of defining this dependence
measure?
15/52
33. Finding covariance with smooth transformations
Illustration: two variables with no correlation but strong dependence.
-2 -1 0 1 2
-1.5
-1
-0.5
0
0.5
1
1.5
Correlation: 0.00
16/52
34. Finding covariance with smooth transformations
Illustration: two variables with no correlation but strong dependence.
-2 -1 0 1 2
-1.5
-1
-0.5
0
0.5
1
1.5
Correlation: 0.00
-2 0 2
-1
-0.5
0
0.5
-2 0 2
-1
-0.5
0
0.5
16/52
36. Define two spaces, one for each witness
Function in p
f @xA a
IX
j =1
fj 'j @xA
Feature map
Kernel for RKHS p on ˆ:
k@x;xHA a h'@xA;'@xHAip
Function in q
g@yA a
IX
j =1
gj j @yA
Feature map
Kernel for RKHS q on ‰:
l@x;xHA a h@yA;@yHAiq
17/52
38. The constrained covariance
The constrained covariance is
COCO@PXY A a sup
kf kp 1
kgkq 1
cov
2
4
0
@
IX
j =1
fj 'j @xA
1
A
0
@
IX
j =1
gj j @yA
1
A
3
5
18/52
39. The constrained covariance
The constrained covariance is
COCO@PXY A a sup
kf kp 1
kgkq 1
Exy
2
4
0
@
IX
j =1
fj 'j @xA
1
A
0
@
IX
j =1
gj j @yA
1
A
3
5
Fine print: feature mappings '(x) and (y) assumed to have zero mean.
18/52
40. The constrained covariance
The constrained covariance is
COCO@PXY A a sup
kf kp 1
kgkq 1
Exy
2
4
0
@
IX
j =1
fj 'j @xA
1
A
0
@
IX
j =1
gj j @yA
1
A
3
5
Fine print: feature mappings '(x) and (y) assumed to have zero mean.
Rewriting:
Exy ‘f @xAg@yA“
a
2
6
6
4
f1
f2
...
3
7
7
5
b
Exy
0
B
B
@
2
6
6
4
'1@xA
'2@xA
...
3
7
7
5
h
1@yA 2@yA :::
i
1
C
C
A
| {z }
C'(x)(y)
2
6
6
4
g1
g2
...
3
7
7
5
18/52
41. The constrained covariance
The constrained covariance is
COCO@PXY A a sup
kf kp 1
kgkq 1
Exy
2
4
0
@
IX
j =1
fj 'j @xA
1
A
0
@
IX
j =1
gj j @yA
1
A
3
5
Fine print: feature mappings '(x) and (y) assumed to have zero mean.
Rewriting:
Exy ‘f @xAg@yA“
a
2
6
6
4
f1
f2
...
3
7
7
5
b
Exy
0
B
B
@
2
6
6
4
'1@xA
'2@xA
...
3
7
7
5
h
1@yA 2@yA :::
i
1
C
C
A
| {z }
C'(x)(y)
2
6
6
4
g1
g2
...
3
7
7
5
COCO: max singular value of feature covariance C'(x)(y)
18/52
42. Computing COCO in practice
Given sample f@xi ;yi Agn
i=1
i:i:d:
$ PXY , what is empirical COCO ?
19/52
43. Computing COCO in practice
Given sample f@xi ;yi Agn
i=1
i:i:d:
$ PXY , what is empirical COCO ?
COCO is largest eigenvalue
max of
0 1
n KL
1
n LK 0
#
45. #
:
Kij a k@xi ;xj A and Lij a l@yi ;yj A.
Fine print: kernels are computed with empirically centered features '(x) 1
n
Pn
i=1
'(xi )
and (y) 1
n
Pn
i=1
(yi ).
AG., A. Smola., O. Bousquet, R. Herbrich, A. Belitski, M. Augath, Y. Murayama, J. Pauls, B.
Schoelkopf, and N. Logothetis, AISTATS’05
19/52
46. Computing COCO in practice
Given sample f@xi ;yi Agn
i=1
i:i:d:
$ PXY , what is empirical COCO ?
COCO is largest eigenvalue
max of
0 1
n KL
1
n LK 0
#
48. #
:
Kij a k@xi ;xj A and Lij a l@yi ;yj A.
Witness functions (singular vectors):
f @xA G
mX
i=1
i k@xi ;xA g@yA G
nX
i=1
49. i l@yi ;yA
Fine print: kernels are computed with empirically centered features '(x) 1
n
Pn
i=1
'(xi )
and (y) 1
n
Pn
i=1
(yi ).
AG., A. Smola., O. Bousquet, R. Herbrich, A. Belitski, M. Augath, Y. Murayama, J. Pauls, B.
Schoelkopf, and N. Logothetis, AISTATS’05
19/52
50. What is a large dependence with COCO?
−2 0 2
−3
−2
−1
0
1
2
3
X
Y
Smooth density
−4 −2 0 2 4
−4
−2
0
2
4
X
Y
500 Samples, smooth density
−2 0 2
−3
−2
−1
0
1
2
3
X
Y
Rough density
−4 −2 0 2 4
−4
−2
0
2
4
X
Y
500 samples, rough density
Density takes the form:
PXY G 1Csin@!xAsin@!yA
Which of these is the more “dependent”?
20/52
57. Dependence largest when at “low” frequencies
As dependence is encoded at higher frequencies, the smooth
mappings f ;g achieve lower linear dependence.
Even for independent variables, COCO will not be zero at finite
sample sizes, since some mild linear dependence will be found by f,g
(bias)
This bias will decrease with increasing sample size.
27/52
58. Can we do better than COCO?
A second example with zero correlation.
First singular value of feature covariance C'(x)(y):
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Correlation: 0.00
-2 0 2
-1
-0.5
0
0.5
-2 0 2
-1
-0.5
0
0.5
-1 -0.5 0 0.5
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Correlation: 0.80 COCO1
: 0.11
28/52
59. Can we do better than COCO?
A second example with zero correlation.
Second singular value of feature covariance C'(x)(y):
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Correlation: 0.00
-2 0 2
-1
-0.5
0
0.5
1
-2 0 2
-1
-0.5
0
0.5
1
28/52
60. Can we do better than COCO?
A second example with zero correlation.
Second singular value of feature covariance C'(x)(y):
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Correlation: 0.00
-2 0 2
-1
-0.5
0
0.5
1
-2 0 2
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-0.5
0
0.5
Correlation: 0.37 COCO2
: 0.06
28/52
61. The Hilbert-Schmidt Independence Criterion
Writing the ith singular value of the feature covariance C'(x)(y) as
i Xa COCOi @PXY Yp;qA;
define Hilbert-Schmidt Independence Criterion (HSIC)
HSIC2
@PXY Yp;qA a
IX
i=1
2
i :
AG, O. Bousquet , A. Smola., and B. Schoelkopf, ALT2005; AG,.,K. Fukumizu„C.H. Teo., L. Song., B.
Schoelkopf., and A. Smola, NIPS 2007,.
29/52
62. The Hilbert-Schmidt Independence Criterion
Writing the ith singular value of the feature covariance C'(x)(y) as
i Xa COCOi @PXY Yp;qA;
define Hilbert-Schmidt Independence Criterion (HSIC)
HSIC2
@PXY Yp;qA a
IX
i=1
2
i :
AG, O. Bousquet , A. Smola., and B. Schoelkopf, ALT2005; AG,.,K. Fukumizu„C.H. Teo., L. Song., B.
Schoelkopf., and A. Smola, NIPS 2007,.
HSIC is MMD with product kernel!
HSIC2
@PXY Yp;qA a MMD2
@PXY ;PX PY YrA
where @@x;yA;@xH;yHAA a k@x;xHAl@y;yHA.
29/52
63. Asymptotics of HSIC under independence
Given sample f@xi ;yi gn
i=1
i:i:d:
$ PXY , what is empirical HSIC?
Empirical HSIC (biased)
HSIC a 1
n2
trace@KLA
Kij a k@xi ;xj A and Lij a l@yi yj A (K and L computed with
empirically centered features)
Statistical testing: given PXY a PX PY , what is the threshold c
such that P@HSIC cA for small ?
Asymptotics of HSIC when PXY a PX PY :
n HSIC
D
3
IX
l=1
l z2
l ; zl $ x@0;1Ai:i:d:
where l l (zj ) =
R
hijqr l (zi )dFi;q;r ; hijqr = 1
4!
P(i;j ;q;r)
(t;u;v;w)
ktu ltu + ktu lvw 2ktu ltv
30/52
64. Asymptotics of HSIC under independence
Given sample f@xi ;yi gn
i=1
i:i:d:
$ PXY , what is empirical HSIC?
Empirical HSIC (biased)
HSIC a 1
n2
trace@KLA
Kij a k@xi ;xj A and Lij a l@yi yj A (K and L computed with
empirically centered features)
Statistical testing: given PXY a PX PY , what is the threshold c
such that P@HSIC cA for small ?
Asymptotics of HSIC when PXY a PX PY :
n HSIC
D
3
IX
l=1
l z2
l ; zl $ x@0;1Ai:i:d:
where l l (zj ) =
R
hijqr l (zi )dFi;q;r ; hijqr = 1
4!
P(i;j ;q;r)
(t;u;v;w)
ktu ltu + ktu lvw 2ktu ltv
30/52
65. Asymptotics of HSIC under independence
Given sample f@xi ;yi gn
i=1
i:i:d:
$ PXY , what is empirical HSIC?
Empirical HSIC (biased)
HSIC a 1
n2
trace@KLA
Kij a k@xi ;xj A and Lij a l@yi yj A (K and L computed with
empirically centered features)
Statistical testing: given PXY a PX PY , what is the threshold c
such that P@HSIC cA for small ?
Asymptotics of HSIC when PXY a PX PY :
n HSIC
D
3
IX
l=1
l z2
l ; zl $ x@0;1Ai:i:d:
where l l (zj ) =
R
hijqr l (zi )dFi;q;r ; hijqr = 1
4!
P(i;j ;q;r)
(t;u;v;w)
ktu ltu + ktu lvw 2ktu ltv
30/52
66. Asymptotics of HSIC under independence
Given sample f@xi ;yi gn
i=1
i:i:d:
$ PXY , what is empirical HSIC?
Empirical HSIC (biased)
HSIC a 1
n2
trace@KLA
Kij a k@xi ;xj A and Lij a l@yi yj A (K and L computed with
empirically centered features)
Statistical testing: given PXY a PX PY , what is the threshold c
such that P@HSIC cA for small ?
Asymptotics of HSIC when PXY a PX PY :
n HSIC
D
3
IX
l=1
l z2
l ; zl $ x@0;1Ai:i:d:
where l l (zj ) =
R
hijqr l (zi )dFi;q;r ; hijqr = 1
4!
P(i;j ;q;r)
(t;u;v;w)
ktu ltu + ktu lvw 2ktu ltv
30/52
67. A statistical test
Given PXY a PX PY , what is the threshold c such that
P@HSIC cA for small (prob. of false positive)?
Original time series:
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10
Permutation:
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
Y7 Y3 Y9 Y2 Y4 Y8 Y5 Y1 Y6 Y10
Null distribution via permutation
Compute HSIC for fxi ;y(i)gn
i=1 for random permutation of indices
f1;:::;ng. This gives HSIC for independent variables.
Repeat for many different permutations, get empirical CDF
Threshold c is 1 quantile of empirical CDF 31/52
68. A statistical test
Given PXY a PX PY , what is the threshold c such that
P@HSIC cA for small (prob. of false positive)?
Original time series:
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10
Permutation:
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
Y7 Y3 Y9 Y2 Y4 Y8 Y5 Y1 Y6 Y10
Null distribution via permutation
Compute HSIC for fxi ;y(i)gn
i=1 for random permutation of indices
f1;:::;ng. This gives HSIC for independent variables.
Repeat for many different permutations, get empirical CDF
Threshold c is 1 quantile of empirical CDF 31/52
69. A statistical test
Given PXY a PX PY , what is the threshold c such that
P@HSIC cA for small (prob. of false positive)?
Original time series:
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10
Permutation:
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
Y7 Y3 Y9 Y2 Y4 Y8 Y5 Y1 Y6 Y10
Null distribution via permutation
Compute HSIC for fxi ;y(i)gn
i=1 for random permutation of indices
f1;:::;ng. This gives HSIC for independent variables.
Repeat for many different permutations, get empirical CDF
Threshold c is 1 quantile of empirical CDF 31/52
70. Application: dependence detection across languages
Testing task: detect dependence between English and French text
Les ordres de gouvernements
provinciaux et municipaux
subissent de fortes pressions
Honourable senators, I have a
question for the Leader of the
Government in the Senate
Text from the aligned hansards of the 36th parliament of canada,
https://www.isi.edu/natural-language/download/hansard/
YX
Honorables sénateurs, ma question
s’adresse au leader du
gouvernement au Sénat
Au contraire, nous avons augmenté
le financement fédéral pour le
développement des jeunes
No doubt there is great pressure
on provincial and municipal
governments
In fact, we have increased
federal investments for early
childhood development.
...
...
32/52
71. Application: dependence detection across languages
Testing task: detect dependence between English and French text
k-spectrum kernel, k a 10, sample size n a 10
HSIC a 1
n2
trace@KLA
(K and L column centered) 33/52
72. Application:Dependence detection across languages
Results (for a 0:05)
k-spectrum kernel: average Type II error 0
Bag of words kernel: average Type II error 0.18
Settings: Five line extracts, averaged over 300 repetitions, for
“Agriculture” transcripts. Similar results for Fisheries and
Immigration transcripts.
34/52
74. Detecting higher order interaction
How to detect V-structures with pairwise weak individual
dependence?
X Y
Z
36/52
75. Detecting higher order interaction
How to detect V-structures with pairwise weak individual
dependence?
36/52
76. Detecting higher order interaction
How to detect V-structures with pairwise weak individual
dependence?
X cc Y ;Y cc Z;X cc Z
X1 vs Y1 Y1 vs Z1
X1 vs Z1 X1*Y1 vs Z1
X Y
Z
X ;Y
i:i:d:
$ x@0;1A
Zj X ;Y $ sign@XY AExp@ 1p
2
A
Fine print: Faithfulness violated here!
36/52
77. V-structure discovery
X Y
Z
Assume X cc Y has been established.
V-structure can then be detected by:
Consistent CI test: H0 X X cc Y jZ [Fukumizu et al. 2008, Zhang et al. 2011]
Factorisation test: H0 X @X ;Y A cc Z • @X ;ZA cc Y • @Y ;ZA cc X
(multiple standard two-variable tests)
How well do these work?
37/52
78. Detecting higher order interaction
Generalise earlier example to p dimensions
X cc Y ;Y cc Z;X cc Z
X1 vs Y1 Y1 vs Z1
X1 vs Z1 X1*Y1 vs Z1
X Y
Z
X ;Y
i:i:d:
$ x@0;1A
Zj X ;Y $ sign@XY AExp@ 1p
2
A
X2:p;Y2:p;Z2:p
i:i:d:
$ x@0;Ip 1A
Fine print: Faithfulness violated here!
38/52
80. Lancaster interaction measure
Lancaster interaction measure of @X1;:::;XD A $ P is a signed
measure ¡P that vanishes whenever P can be factorised non-trivially.
D a 2 X ¡LP a PXY PX PY
40/52
81. Lancaster interaction measure
Lancaster interaction measure of @X1;:::;XD A $ P is a signed
measure ¡P that vanishes whenever P can be factorised non-trivially.
D a 2 X ¡LP a PXY PX PY
D a 3 X ¡LP a PXYZ PX PYZ PY PXZ PZ PXY C2PX PY PZ
40/52
82. Lancaster interaction measure
Lancaster interaction measure of @X1;:::;XD A $ P is a signed
measure ¡P that vanishes whenever P can be factorised non-trivially.
D a 2 X ¡LP a PXY PX PY
D a 3 X ¡LP a PXYZ PX PYZ PY PXZ PZ PXY C2PX PY PZ
X Y
Z
X Y
Z
X Y
Z
X Y
Z
PXY Z −PXPY Z −PY PXZ −PZPXY +2PXPY PZ
∆LP =
40/52
83. Lancaster interaction measure
Lancaster interaction measure of @X1;:::;XD A $ P is a signed
measure ¡P that vanishes whenever P can be factorised non-trivially.
D a 2 X ¡LP a PXY PX PY
D a 3 X ¡LP a PXYZ PX PYZ PY PXZ PZ PXY C2PX PY PZ
X Y
Z
X Y
Z
X Y
Z
X Y
Z
PXY Z −PXPY Z −PXZPY −PXY PZ +2PXPY PZ
∆LP = 0
Case of PX cc PYZ
40/52
84. Lancaster interaction measure
Lancaster interaction measure of @X1;:::;XD A $ P is a signed
measure ¡P that vanishes whenever P can be factorised non-trivially.
D a 2 X ¡LP a PXY PX PY
D a 3 X ¡LP a PXYZ PX PYZ PY PXZ PZ PXY C2PX PY PZ
@X ;Y A cc Z • @X ;ZA cc Y • @Y ;ZA cc X A ¡LP a 0:
...so what might be missed?
40/52
85. Lancaster interaction measure
Lancaster interaction measure of @X1;:::;XD A $ P is a signed
measure ¡P that vanishes whenever P can be factorised non-trivially.
D a 2 X ¡LP a PXY PX PY
D a 3 X ¡LP a PXYZ PX PYZ PY PXZ PZ PXY C2PX PY PZ
¡LP a 0 ;@X ;Y A cc Z • @X ;ZA cc Y • @Y ;ZA cc X
Example:
P(0;0;0) = 0:2 P(0;0;1) = 0:1 P(1;0;0) = 0:1 P(1;0;1) = 0:1
P(0;1;0) = 0:1 P(0;1;1) = 0:1 P(1;1;0) = 0:1 P(1;1;1) = 0:2
40/52
86. A kernel test statistic using Lancaster Measure
Construct a test by estimating k @¡LPAk2
r ; where a k l m:
k@PXYZ PXY PZ ¡¡¡Ak2
r a
hPXYZ ;PXYZ ir 2 hPXYZ ;PXY PZ ir ¡¡¡
41/52
87. A kernel test statistic using Lancaster Measure
Table: V -statistic estimators of h;Hir
(without terms PX PY PZ ). H
is centering matrix I n 1
Lancaster interaction statistic: D. Sejdinovic, AG, W. Bergsma, NIPS13
k @¡LPAk2
r a 1
n2
@HKH HLH HMHA++ :
Empirical joint central moment in the feature space
42/52
88. A kernel test statistic using Lancaster Measure
Table: V -statistic estimators of h;Hir
(without terms PX PY PZ ). H
is centering matrix I n 1
Lancaster interaction statistic: D. Sejdinovic, AG, W. Bergsma, NIPS13
k @¡LPAk2
r a 1
n2
@HKH HLH HMHA++ :
Empirical joint central moment in the feature space
42/52
90. Interaction for D 4
Interaction measure valid for all D:
(Streitberg, 1990)
¡S P a
X
@ 1Ajj 1
@jj 1A3JP
For a partition , J associates to the
joint the corresponding factorisation,
e.g., J13j2j4P = PX1X3
PX2
PX4
:
44/52
91. Interaction for D 4
Interaction measure valid for all D:
(Streitberg, 1990)
¡S P a
X
@ 1Ajj 1
@jj 1A3JP
For a partition , J associates to the
joint the corresponding factorisation,
e.g., J13j2j4P = PX1X3
PX2
PX4
:
44/52
92. Interaction for D 4
Interaction measure valid for all D:
(Streitberg, 1990)
¡S P a
X
@ 1Ajj 1
@jj 1A3JP
For a partition , J associates to the
joint the corresponding factorisation,
e.g., J13j2j4P = PX1X3
PX2
PX4
:
1e+04
1e+09
1e+14
1e+19
1 3 5 7 9 11 13 15 17 19 21 23 25
D
Numberofpartitionsof{1,...,D}
Bell numbers growth
44/52