With the increasing use of AI and ML-based systems, interpretability is becoming an increasingly important issue to ensure user trust and safety. This also applies to the area of recommender systems, where methods based on matrix factorization (MF) are among the most popular methods for collaborative filtering tasks with implicit feedback. Despite their simplicity, the latent factors of users and items lack interpretability in the case of the effective, unconstrained MF-based methods. In this work, we propose an extended latent Dirichlet Allocation model (LDAext) that has interpretable parameters such as user cohorts of item preferences and the affiliation of a user with different cohorts. We prove a theorem on how to transform the factors of an unconstrained MF model into the parameters of LDAext. Using this theoretical connection, we train an MF model on different real-world data sets, transform the latent factors into the parameters of LDAext and test their interpretation in several experiments for plausibility. Our experiments confirm the interpretability of the transformed parameters and thus demonstrate the usefulness of our proposed approach.

1. An Interpretable Model for Collaborative
Filtering Using an Extended Latent
Dirichlet Allocation Approach
Florian Wilhelm, Marisa Mohr and Lien Michiels

2. Motivation
Matrix factorization based methods for collaborative
filtering tasks still show the best performance in various
benchmarks.
2
Anelli et al. 2021. Reenvisioning the comparison between Neural Collaborative Filtering and Matrix Factorization. DOI:https://doi.org/10.1145/3460231.3475944
Rendle et al. 2020. Neural Collaborative Filtering vs. Matrix Factorization Revisited. DOI:https://doi.org/10.1145/3383313.3412488
Dacrema et al. 2019. Are we really making much progress? A worrying analysis of recent neural recommendation approaches. DOI:https://doi.org/10.1145/3298689.3347058
But how can we interpret the results?

3. Matrix Factorization
where
with set of users , items and latent dimension .
induces a personalized ranking .
3
X ⇡ X̂ := WHt
,
I
U
x̂ui = hwu, hii + bi
X 2 R|U|⇥|I|
, W 2 R|U|⇥|K|
, H 2 R|I|⇥|K|
K
>u

4. (Semi) Non-negative Matrix Factorization
Matrix Factorization
is Semi Non-negative (SNMF) when
and Non-negative (NMF) when
4
x̂ui = hwu, hii + bi
<latexit sha1_base64="w9kn1lrcQuyPGBuvaQyBbnM/5+E=">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</latexit>
wu 2 R
|K|
0 , hi 2 R
|K|
0 , bi 2 R 0.
<latexit sha1_base64="m3zfLS4yC3sv1XYR6glDnl3K4k4=">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</latexit>
wu 2 R|K|
, hi 2 R
|K|
0 , bi 2 R 0

5. Research Questions
1. How can the factors and of the matrix
factorization be interpreted?
2. Are the interpretations plausible on real data sets?
5
W H
Interpret matrix factorization as a
Latent Dirichlet Allocation problem.

6. Classical Latent Dirichlet Allocation Model
6
|S|
|U|
|K|
Æ µu zus ius
Ø 'k
1. Choose
2. Choose
3. For user and interaction :
a) Choose cohort
b) Choose item
✓u ⇠ Dirichlet(↵).
'k ⇠ Dirichlet( ).
zus ⇠ Categorical(✓u).
ius ⇠ p(ius|'zus
) :=
Categorical('zus
).
u z

7. Shortcomings of Classical LDA for RecSys
1. Item preferences only depend on the user cohorts since
no explicit item popularity is included.
2. If existed, there would be no way of weighting the
item preferences of the cohort against the item
popularities for a user.
7
bi
bi
'k
bi
Matrix factorization does not
have those shortcomings.

8. LDAext Model
8
Extends classical LDA with
item popularity and
user conformity .
Item probability:
i
u
ius ⇠ p(ius|'zus
, i, u) :=
Categorical(kck1
1
c)
c = 'zus
+ u ·
with

9. Theorem: Reformulation of MF as LDAext
9
<latexit sha1_base64="szfKBmHyhyvY2k0rQITadQ2iqBo=">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</latexit>
x̂ui = hwu, hii + bi
<latexit sha1_base64="3MUkxgrMutWb2JBEhIM7DQGFIwA=">AAAB9XicbVDLTgJBEJzFF+IL9ehlIph4IrvEqEeiF4+YyCOBlcwODUyYnV1nejWE8B9ePGiMV//Fm3/jAHtQsJJOKlXd6e4KYikMuu63k1lZXVvfyG7mtrZ3dvfy+wd1EyWaQ41HMtLNgBmQQkENBUpoxhpYGEhoBMPrqd94BG1EpO5wFIMfsr4SPcEZWum+2O7Dg5FMYScp0k6+4JbcGegy8VJSICmqnfxXuxvxJASFXDJjWp4boz9mGgWXMMm1EwMx40PWh5alioVg/PHs6gk9sUqX9iJtSyGdqb8nxiw0ZhQGtjNkODCL3lT8z2sl2Lv0x0LFCYLi80W9RFKM6DQC2hUaOMqRJYxrYW+lfMA042iDytkQvMWXl0m9XPLOS2e35ULlKo0jS47IMTklHrkgFXJDqqRGONHkmbySN+fJeXHenY95a8ZJZw7JHzifP7spkgQ=</latexit>
>u
<latexit sha1_base64="pGjAJ0TSBZTxR76lKUFgRCb51t8=">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</latexit>
x0
ui = h✓u, p⇤i(u)i
<latexit sha1_base64="lDcWEFigMci83z6/ORrQLNdJqTY=">AAACP3icbVA9SwNBEN3z2/gVtbRZTAQLCXdB1FK0sVQwMZCEY25vkizZ+2B3TghH/pmNf8HO1sZCEVs7NzGFJg4s+3hvHjPzglRJQ6777MzNLywuLa+sFtbWNza3its7dZNkWmBNJCrRjQAMKhljjSQpbKQaIQoU3gX9y5F+d4/ayCS+pUGK7Qi6sexIAWQpv1gvt4JEhWYQ2S9vUQ8Jhn5WPuJ/hXvQaU8O/f6MEqKyljGt7NwQrJv7xZJbccfFZ4E3ASU2qWu/+NQKE5FFGJNQYEzTc1Nq56BJCoXDQiszmILoQxebFsYQoWnn4/uH/MAyIe8k2r6Y+Jj97cghMqN1bWcE1DPT2oj8T2tm1Dlr5zJOM8JY/AzqZIpTwkdh8lBqFKQGFoDQ0u7KRQ80CLKRF2wI3vTJs6BerXgnleObaun8YhLHCttj++yQeeyUnbMrds1qTLAH9sLe2Lvz6Lw6H87nT+ucM/Hssj/lfH0DhQWvwA==</latexit>
✓u, 'k, , u
Total Ranking
Determine

10. Sketch of Proof
10
<latexit sha1_base64="bVu1Kan+29pOhGjp8DxvhCDbXlI=">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</latexit>
Define non-negative user vectors w0
u = [w+
u, w u] where
w+
uk =
(
wuk if wuk 0
0 otherwise
and w uk =
(
wuk if wuk < 0
0 otherwise
,
and item vectors h0
i = [hi + s, hi + s] with s = (si)i2I, si = maxk2K |hik|
Define non-negative user vectors where
and item vectors with
and
otherwise otherwise
if if
Step 1: Transformation of MF into NMF

11. Sketch of Proof
11
Step 2: Rearrange to have categorical distributions
<latexit sha1_base64="wRKB4vMsCAqHdwj5meQk7tZyINA=">AAADSnicdVLPb9MwFHayAVv5sQ6OXCw6EIdSJRMCLpMmuHAcEu0m1SW8OG5j1XEi29lUWf77uHDixh/BhQMT4oKTtRK025OSfH7v8+cvzy+tBNcmir4H4db2rdt3dnY7d+/df7DX3X840mWtKBvSUpTqLAXNBJdsaLgR7KxSDIpUsNN0/q6pn54zpXkpP5pFxSYFzCSfcgrGp5L94PMBSdmMSwtKwcJZIVwHk7QUmV4U/mPJOagq5y6Z4yMyYspgUoDJ06nNXWIJaIPnDreVJP5kX8TuOkIfP8NruhkTBty6aHqDVuolCPESJmcGElv7Q48wESBngpEcjF0xL5xL6v5qJf2CqJbVCpJ+S75wrYRs395cp5Eq0gySemVoQ3PN2A22G5eEyWzZ0IOk24sGURt4E8RL0EPLOEm630hW0rpg0lABWo/jqDITL2c4Fcx1SK1ZBXQOMzb2UELB9MS2o+DwU5/J8LRU/pEGt9l/d1godHMBntk41+u1JnldbVyb6ZuJ5bKqDZP06qBpLbApcTNXOOOKUSMWHgBV3HvFNAcF1Pjp6/gmxOu/vAlGh4P41eDlh8Pe8dtlO3bQY/QEPUcxeo2O0Xt0goaIBl+CH8Gv4DL8Gv4Mf4d/rqhhsNzzCP0XW9t/Af8NEvw=</latexit>
'k = kh⇤kk 1
1 h⇤k, = kbk 1
1 b,
✓uk = hŵu, nui 1
ŵuknuk, u = kŵuk 1
1 kbk1
<latexit sha1_base64="n0SadQlLWm/Ll734szoV0D9pjXU=">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</latexit>
'k = kh⇤kk 1
1 h⇤k, = kbk 1
1 b,
✓uk = hŵu, nui 1
ŵuknuk, u = kŵuk 1
1 k
where ŵuk = kh⇤kk1wuk and nuk =
P
i2I 'ki + u i.
and
where
<latexit sha1_base64="d4WBG9e+wecT/YS5KSsxJ+slqbA=">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</latexit>
x0
ui = h✓u, p⇤i(u)i = . . . = hŵu, nui 1
x̂ui

12. Hypothesis Tests on Real Data Sets
LDAext has an inherent interpretability of all variables
but do they also have a practical meaning?
12

13. Experiments
13
1. Cohort Allocation: Meaning of , do users interact
more with items from the cohort they prefer the most?
2. Popularity Ranking: How strong is correlated with
the empirical item popularity?
3. Conformity Ranking: How strong is correlated with
the average popularity of the interacted items .
4. User’s Preferences: Is the overlap a good
proxy for the Jaccard index and thus users
can be compared using the preferences .
<latexit sha1_base64="yhLdTXRu2LCCOWwIS+Bc1/APLmQ=">AAAB+nicbVBLSwMxGMzWV11fWz16CRbBU9kVUS9i0YvHCvYB7VKy2Wwbmk2WJKuUtT/Fi4IiXv0P3r2I/8Zs24O2DoQMM99HJhMkjCrtut9WYWFxaXmluGqvrW9sbjml7YYSqcSkjgUTshUgRRjlpK6pZqSVSILigJFmMLjM/eYtkYoKfqOHCfFj1OM0ohhpI3WdUicQLFTD2FydkDCNuk7ZrbhjwHniTUn5/MM+S56+7FrX+eyEAqcx4RozpFTbcxPtZ0hqihkZ2Z1UkQThAeqRtqEcxUT52Tj6CO4bJYSRkOZwDcfq740MxSpPZyZjpPtq1svF/7x2qqNTP6M8STXhePJQlDKoBcx7gCGVBGs2NARhSU1WiPtIIqxNW7YpwZv98jxpHFa848rRtVuuXoAJimAX7IED4IETUAVXoAbqAIM78ACewYt1bz1ar9bbZLRgTXd2wB9Y7z8j25eU</latexit>
<latexit sha1_base64="UiP45uQE+CxmKaweDOuD33lucug=">AAAB8HicbVDLSgMxFL3js9ZXVXDjJlgEV2VGRF2WunHZgn1IO5RMJtOGJpkhyQhl6Fe4caGIW8G/8AvcufFbTB8LbT0QOJxzLrn3BAln2rjul7O0vLK6tp7byG9ube/sFvb2GzpOFaF1EvNYtQKsKWeS1g0znLYSRbEIOG0Gg+ux37ynSrNY3pphQn2Be5JFjGBjpbsOt9EQd9NuoeiW3AnQIvFmpFg+rH2z98pHtVv47IQxSQWVhnCsddtzE+NnWBlGOB3lO6mmCSYD3KNtSyUWVPvZZOEROrFKiKJY2ScNmqi/JzIstB6KwCYFNn09743F/7x2aqIrP2MySQ2VZPpRlHJkYjS+HoVMUWL40BJMFLO7ItLHChNjO8rbErz5kxdJ46zkXZTOa7aNCkyRgyM4hlPw4BLKcANVqAMBAQ/wBM+Och6dF+d1Gl1yZjMH8AfO2w+vdZQL</latexit>
u <latexit sha1_base64="NuPTYjnLix0O4m068GEu8zHF4Uc=">AAAB+HicbVDLSsNAFJ34rPXRqODGzWARXJVERF2WutFdC/YBbQiT6aQdOpmEeQg15EvcuFDEhRu/wS9w58ZvcdJ2oa0HBg7n3Ms9c4KEUakc58taWl5ZXVsvbBQ3t7Z3SvbuXkvGWmDSxDGLRSdAkjDKSVNRxUgnEQRFASPtYHSV++07IiSN+a0aJ8SL0IDTkGKkjOTbpV6E1BAjlt5kfqoz3y47FWcCuEjcGSlXDxrf9K32Ufftz14/xjoiXGGGpOy6TqK8FAlFMSNZsaclSRAeoQHpGspRRKSXToJn8NgofRjGwjyu4ET9vZGiSMpxFJjJPKac93LxP6+rVXjppZQnWhGOp4dCzaCKYd4C7FNBsGJjQxAW1GSFeIgEwsp0VTQluPNfXiSt04p7XjlrmDZqYIoCOARH4AS44AJUwTWogybAQIMH8ASerXvr0XqxXqejS9ZsZx/8gfX+Azd9lyo=</latexit>
Iu
<latexit sha1_base64="n/N+YvD4JPepnQGST65E+AHvZhU=">AAACAHicbVDLSsNAFJ3UV62vqKALN8EiuCqJiLosdeOyBfuAJoTJZNIOnTyYuRFKyMZfcePCIm5d+wXu3PgtTtoutPXAMIdz7uXee7yEMwmm+aWVVlbX1jfKm5Wt7Z3dPX3/oCPjVBDaJjGPRc/DknIW0TYw4LSXCIpDj9OuN7ot/O4DFZLF0T2ME+qEeBCxgBEMSnL1Y9uLuS/HofoyG4YUcO5mae7qVbNmTmEsE2tOqvWj1jebND6arv5p+zFJQxoB4VjKvmUm4GRYACOc5hU7lTTBZIQHtK9ohEMqnWx6QG6cKcU3glioF4ExVX93ZDiUxY6qMsQwlIteIf7n9VMIbpyMRUkKNCKzQUHKDYiNIg3DZ4IS4GNFMBFM7WqQIRaYgMqsokKwFk9eJp2LmnVVu2ypNBpohjI6QafoHFnoGtXRHWqiNiIoR0/oBU20R+1Ze9XeZqUlbd5ziP5Ae/8B9xSbAA==</latexit>
✓u
<latexit sha1_base64="fn4ExXU17TtrK25PdggSkvTGeT4=">AAACHXicbVDLSgMxFM3UV62vUZdugq1YQcpMKeqy6MadFewD2jJkMmkbmnmQ3BHKMD/ixl9x40IRF27EvzF9LLTtgZDDOfdy7z1uJLgCy/oxMiura+sb2c3c1vbO7p65f9BQYSwpq9NQhLLlEsUED1gdOAjWiiQjvitY0x3ejP3mI5OKh8EDjCLW9Uk/4D1OCWjJMSuFu2LHDYWnRr7+kg4MGJDUic/xMjmJT9OzgmPmrZI1AV4k9ozk0Qw1x/zqeCGNfRYAFUSptm1F0E2IBE4FS3OdWLGI0CHps7amAfGZ6iaT61J8ohUP90KpXwB4ov7tSIivxmvqSp/AQM17Y3GZ146hd9VNeBDFwAI6HdSLBYYQj6PCHpeMghhpQqjkeldMB0QSCjrQnA7Bnj95kTTKJfuiVLkv56vXsziy6AgdoyKy0SWqoltUQ3VE0RN6QW/o3Xg2Xo0P43NamjFmPYfoH4zvX08Aoq8=</latexit>
O(✓u, ✓u0 )
<latexit sha1_base64="/NyEKRWXelZ716dnmyTux64DtV4=">AAACDXicbVDLSsNAFJ3UV62vqEs3g61YQUpSRF0W3airCvYBbQiT6aQdOpmEmYlQQn7Ajb/ixoUibt2782+ctF3U1gMDZ865l3vv8SJGpbKsHyO3tLyyupZfL2xsbm3vmLt7TRnGApMGDlko2h6ShFFOGooqRtqRICjwGGl5w+vMbz0SIWnIH9QoIk6A+pz6FCOlJdcsle7K3QCpAUYsuU3d+BTOfpP4OD0puWbRqlhjwEViT0kRTFF3ze9uL8RxQLjCDEnZsa1IOQkSimJG0kI3liRCeIj6pKMpRwGRTjK+JoVHWulBPxT6cQXH6mxHggIpR4GnK7NN5byXif95nVj5l05CeRQrwvFkkB8zqEKYRQN7VBCs2EgThAXVu0I8QAJhpQMs6BDs+ZMXSbNasc8rZ/fVYu1qGkceHIBDUAY2uAA1cAPqoAEweAIv4A28G8/Gq/FhfE5Kc8a0Zx/8gfH1CyfAmv4=</latexit>
J(Iu, Iu0 ) <latexit sha1_base64="P6/GYnHJDoevsbkcAYOc24/KDUc=">AAAB/nicbVDLSsNAFJ3UV62vqOjGTbAIrkoioi5L3bhswT6gCWEymbZDJ5kwcyOUUPBX3LhQils/wC9w58ZvcdJ2oa0Hhjmccy9z5gQJZwps+8sorKyurW8UN0tb2zu7e+b+QUuJVBLaJIIL2QmwopzFtAkMOO0kkuIo4LQdDG9zv/1ApWIivodRQr0I92PWYwSDlnzzyA0ED9Uo0lfmwoACHvupb5btij2FtUycOSlXjxvfbFL7qPvmpxsKkkY0BsKxUl3HTsDLsARGOB2X3FTRBJMh7tOupjGOqPKyafyxdaaV0OoJqU8M1lT9vZHhSOUJ9WSEYaAWvVz8z+um0LvxMhYnKdCYzB7qpdwCYeVdWCGTlAAfaYKJZDqrRQZYYgK6sZIuwVn88jJpXVScq8plQ7dRQzMU0Qk6RefIQdeoiu5QHTURQRl6Qi/o1Xg0no2J8TYbLRjznUP0B8b7DyKqmfQ=</latexit>
✓u

14. Results Cohort Allocation
14
<latexit sha1_base64="86pXu3nRdaaxkOaPEXe8qeFVkwo=">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</latexit>
dataset train test
MovieLens-1M 0.608 ± 0.022 0.162 ± 0.018
Amazon 0.125 ± 0.004 0.059 ± 0.004
Goodbooks 0.178 ± 0.010 0.047 ± 0.005
<latexit sha1_base64="0pDnL/9SXgZtLtglezgKuFDlrp0=">AAACB3icbVBLTgJBEO3xi/hDXbrpCEZXZIYYdUlk4xIT+SQDIT09NdChp3vS3WNCCAfwAG71CO6MW4/hCbyGDcxCwJdU8vJeVarqBQln2rjut7O2vrG5tZ3bye/u7R8cFo6Om1qmikKDSi5VOyAaOBPQMMxwaCcKSBxwaAXD2tRvPYHSTIpHM0qgG5O+YBGjxFjJr8kBiAuNS2GpVyi6ZXcGvEq8jBRRhnqv8NMJJU1jEIZyorXvuYnpjokyjHKY5DuphoTQIemDb6kgMejueHbyBJ9bJcSRVLaEwTP178SYxFqP4sB2xsQM9LI3Ff/z/NREt90xE0lqQND5oijl2Eg8/R+HTAE1fGQJoYrZWzEdEEWosSktbAmkHBoS6IlNxlvOYZU0K2Xvunz1UClW77KMcugUnaFL5KEbVEX3qI4aiCKJXtArenOenXfnw/mct6452cwJWoDz9QsDIZmC</latexit>
Cohen’s d

15. Results Popularity & Conformity Ranking
15
<latexit sha1_base64="+j1StLBWlS4PFxiBhCnQx1LtNkI=">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</latexit>
dataset popularity conformity
MovieLens-1M 0.520 ± 0.006 0.3749 ± 0.0090
Amazon 0.377 ± 0.003 0.0818 ± 0.0026
Goodbooks 0.265 ± 0.004 0.0016 ± 0.0013
<latexit sha1_base64="Mr39HiCdxvqQQmRywrkrMJBOI6s=">AAACG3icbVDLSgNBEJz1bXxFPYowGAVPYTeIehRzEbxEMCokIfROenXI7Mwy0yuEkJPf4Qd41U/wJl49+AX+hpOYg6+CgaKqi+6pOFPSURi+BxOTU9Mzs3PzhYXFpeWV4urahTO5FVgXRhl7FYNDJTXWSZLCq8wipLHCy7hbHfqXt2idNPqcehm2UrjWMpECyEvt4uYp6g4oxbebBHm7us2FwcT7EjUV2sVSWA5H4H9JNCYlNkatXfxodozIUx8WCpxrRGFGrT5YkkLhoNDMHWYgunCNDU81pOha/dE3BnzHKx2eGOufJj5Svyf6kDrXS2M/mQLduN/eUPzPa+SUHLb6Umc5oRZfi5JccTJ82AnvSIuCVM8TEFb6W7m4AQuCfHM/tsTGdAliN/DNRL97+EsuKuVov7x3VikdHY87mmMbbIvtsogdsCN2wmqszgS7Yw/skT0F98Fz8BK8fo1OBOPMOvuB4O0TYOShKA==</latexit>
Kendall ⌧C coefficient

16. Results User’s Preferences
16
<latexit sha1_base64="846tFCs9g0UPtoAMtT2KuN/QJzk=">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</latexit>
dataset other train test
MovieLens-1M bad 1.88 ± 0.03 0.61 ± 0.02
Amazon bad 1.48 ± 0.03 0.86 ± 0.03
Goodbooks bad 2.88 ± 0.06 0.62 ± 0.04
MovieLens-1M rnd 1.17 ± 0.01 0.45 ± 0.02
Amazon rnd 1.44 ± 0.03 0.84 ± 0.03
Goodbooks rnd 2.12 ± 0.04 0.43 ± 0.02
<latexit sha1_base64="0pDnL/9SXgZtLtglezgKuFDlrp0=">AAACB3icbVBLTgJBEO3xi/hDXbrpCEZXZIYYdUlk4xIT+SQDIT09NdChp3vS3WNCCAfwAG71CO6MW4/hCbyGDcxCwJdU8vJeVarqBQln2rjut7O2vrG5tZ3bye/u7R8cFo6Om1qmikKDSi5VOyAaOBPQMMxwaCcKSBxwaAXD2tRvPYHSTIpHM0qgG5O+YBGjxFjJr8kBiAuNS2GpVyi6ZXcGvEq8jBRRhnqv8NMJJU1jEIZyorXvuYnpjokyjHKY5DuphoTQIemDb6kgMejueHbyBJ9bJcSRVLaEwTP178SYxFqP4sB2xsQM9LI3Ff/z/NREt90xE0lqQND5oijl2Eg8/R+HTAE1fGQJoYrZWzEdEEWosSktbAmkHBoS6IlNxlvOYZU0K2Xvunz1UClW77KMcugUnaFL5KEbVEX3qI4aiCKJXtArenOenXfnw/mct6452cwJWoDz9QsDIZmC</latexit>
Cohen’s d

17. Conclusion
1. MF is equivalent to LDAext, which is a plausible model
for the actual dynamics.
2. The factors and can be interpreted using LDAext
with plausible results on real data sets.
17
W H

18. Thank you!
Florian Wilhelm
Head of Data Science
inovex GmbH
Schanzenstraße 6-20
Kupferhütte 1.13
51063 Cologne
Germany
florian.wilhelm@inovex.de

19. Index of Dissimilarity & Overlap
19
<latexit sha1_base64="CYIGJFj8jcr4CHQk7s+hxtDyXQs=">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</latexit>
The index of dissimilarity is defined as
D(u, v) =
1
2
k✓u ✓vk1 2 [0, 1],
and analogously the overlap, i.e.,
O(u, v) = 1 D(u, v) =
X
k2K
min(✓uk, ✓vk) 2 [0, 1].