Missing Data Imputation using Optimal Transport Boris Muzellec Julie - - PowerPoint PPT Presentation

missing data imputation using optimal transport
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Missing Data Imputation using Optimal Transport Boris Muzellec Julie - - PowerPoint PPT Presentation

Missing Data Imputation using Optimal Transport Boris Muzellec Julie Josse Claire Boyer Marco Cuturi <latexit


slide-1
SLIDE 1

Missing Data Imputation using Optimal Transport

Boris Muzellec Julie Josse Claire Boyer Marco Cuturi

slide-2
SLIDE 2

The missing data issue

  • What to do?

Option 1: Remove entries with missing values

information loss, not sustainable

= ⇒

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Option 2: Impute with reasonable guesses

2d 3d 6d 10d Example with 25% missing rate:

With 1% missing rate: 5d: 95% rows kept 300d: 5% rows kept

  • Big data is plagued with missing values
slide-3
SLIDE 3

Outline

  • 1. Missing data and Optimal Transport
  • 2. Non-parametric imputation with OT
  • 3. Fitting parametric imputation models with OT
slide-4
SLIDE 4

How to impute?

  • Mean imputation
  • Regression (conditional expectation)
  • Assuming a joint model:
  • EM + Gaussian distribution: Amelia (Honacker et al., 2011)
  • Low-rank models: Softimpute (Mazumder et al., 2010)
  • VAE and GAN: MIWAE (Mattei & Frellsen, 2019), GAIN (Yoon et al., 2018)
  • Using a conditional model:
  • With logistic, multinomial, Poisson regressions: R’s mice (Van Buuren, 2011)

Deforms joint and marginal distributions This work: Parametric assumption not necessary Preserves distributions Preserves distributions

slide-5
SLIDE 5

Imputing to preserve batch distributions

  • What divergence should we use ?

Wishlist

  • Handles disjoint supports
  • Differentiable
  • Affordable computing times

D ⇣

' 0

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X(imp)

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X(obs)

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  • Contribution:

distribution(X(imp)) ' distribution(X(obs))

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Parametric assumption not necessary

Two batches from the same dataset should have similar distributions. Measure this with a divergence:

slide-6
SLIDE 6

Optimal Transport

MXY = [kXi Y jk2]ij 2 Rn×m

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β = 1 m

m

X

j=1

δY j

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α = 1 n

n

X

i=1

δXi

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OT(α, β)

def

= min

P∈Rn×m

+

P = /n, PT = /m

hP, MXY i

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  • Find the most efficient way of transporting distributions, according to a ground cost
  • Defines a distance for probability distributions
slide-7
SLIDE 7

Optimal Transport

MXY = [kXi Y jk2]ij 2 Rn×m

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β = 1 m

m

X

j=1

δY j

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α = 1 n

n

X

i=1

δXi

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OT(α, β)

def

= min

P∈Rn×m

+

P = /n, PT = /m

hP, MXY i

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  • Find the most efficient way of transporting distributions, according to a ground cost
  • Defines a distance for probability distributions

Handles disjoint supports

slide-8
SLIDE 8

Optimal Transport

MXY = [kXi Y jk2]ij 2 Rn×m

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β = 1 m

m

X

j=1

δY j

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α = 1 n

n

X

i=1

δXi

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Not differentiable Costly:

O(n3 log n)

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OT(α, β)

def

= min

P∈Rn×m

+

P = /n, PT = /m

hP, MXY i

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  • Find the most efficient way of transporting distributions, according to a ground cost
  • Defines a distance for probability distributions

Handles disjoint supports

slide-9
SLIDE 9

Regularized Optimal Transport

MXY = [kXi Y jk2]ij 2 Rn×m

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β = 1 m

m

X

j=1

δY j

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α = 1 n

n

X

i=1

δXi

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Sε(α, β)

def

= OTε(α, β) − 1 2(OTε(α, α) + OTε(β, β))

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OTε(α, β)

def

= min

P∈Rn×m

+

P = /n, PT = /m

hP, MXY i + ε X

ij

pij log pij

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  • Sinkhorn divergence:

(Cuturi, 2013)

slide-10
SLIDE 10

Regularized Optimal Transport

MXY = [kXi Y jk2]ij 2 Rn×m

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β = 1 m

m

X

j=1

δY j

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α = 1 n

n

X

i=1

δXi

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Sε(α, β)

def

= OTε(α, β) − 1 2(OTε(α, α) + OTε(β, β))

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OTε(α, β)

def

= min

P∈Rn×m

+

P = /n, PT = /m

hP, MXY i + ε X

ij

pij log pij

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  • Sinkhorn divergence:

Handles disjoint supports Differentiable Fast computation with Sinkhorn’s algorithm (Cuturi, 2013)

slide-11
SLIDE 11

Imputation Algorithm

  • Input:
  • Initial imputations:

(column mean of observed values + noise)

x(imp)

ij

= x(obs)

:j

+ ε if mij = 1

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for t = 1, 2..., T

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  • Sample batch with no missing values:

X(obs)

i1,...,iK

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Sample batch with missing values:

X(imp)

j1,...,jK

<latexit sha1_base64="m6jWlyPjK0lbKr5sNym3hgIC48=">ACXicbZDLSsNAGIUn9VbrerSzWARKpSQiKDLohvBTQV7gTaGyXTSTjtJhplJsYQ8gPgw4q69QFc+zZOahbaelZn/vMN/P/xOKNSWdaXUVhZXVvfKG6WtrZ3dvfK+wctGcUCkyaOWCQ6HpKE0ZA0FVWMdLgKPAYaXvj6yxvT4iQNArv1ZQTJ0CDkPoUI6VHbrnSC5Aaen7SR+SKg34aeomI9euQdM0a3Dk3qYlTVmNRdcNnZuKiBXwy1/9voRjgMSKsyQlF3b4spJkFAUM5KWerEkHOExGpBuf0K5DFApJM8zu9J4YkfCaiGBM7fv/EBVJOA08z2d5yMcuG/2XdWPmXTkJDHisSYo3ozI8ZVBHMaoF9KghWbKoNwoLqRSEeIoGw0uVlDdiL9y6b1plpW6Z9d16pX+VdFMEROAZVYIMLUAc3oAGaAIMn8AJm4M14Nl6NmfH+gxaM/M8h+CPj4xsTr5id</latexit><latexit sha1_base64="m6jWlyPjK0lbKr5sNym3hgIC48=">ACXicbZDLSsNAGIUn9VbrerSzWARKpSQiKDLohvBTQV7gTaGyXTSTjtJhplJsYQ8gPgw4q69QFc+zZOahbaelZn/vMN/P/xOKNSWdaXUVhZXVvfKG6WtrZ3dvfK+wctGcUCkyaOWCQ6HpKE0ZA0FVWMdLgKPAYaXvj6yxvT4iQNArv1ZQTJ0CDkPoUI6VHbrnSC5Aaen7SR+SKg34aeomI9euQdM0a3Dk3qYlTVmNRdcNnZuKiBXwy1/9voRjgMSKsyQlF3b4spJkFAUM5KWerEkHOExGpBuf0K5DFApJM8zu9J4YkfCaiGBM7fv/EBVJOA08z2d5yMcuG/2XdWPmXTkJDHisSYo3ozI8ZVBHMaoF9KghWbKoNwoLqRSEeIoGw0uVlDdiL9y6b1plpW6Z9d16pX+VdFMEROAZVYIMLUAc3oAGaAIMn8AJm4M14Nl6NmfH+gxaM/M8h+CPj4xsTr5id</latexit><latexit sha1_base64="m6jWlyPjK0lbKr5sNym3hgIC48=">ACXicbZDLSsNAGIUn9VbrerSzWARKpSQiKDLohvBTQV7gTaGyXTSTjtJhplJsYQ8gPgw4q69QFc+zZOahbaelZn/vMN/P/xOKNSWdaXUVhZXVvfKG6WtrZ3dvfK+wctGcUCkyaOWCQ6HpKE0ZA0FVWMdLgKPAYaXvj6yxvT4iQNArv1ZQTJ0CDkPoUI6VHbrnSC5Aaen7SR+SKg34aeomI9euQdM0a3Dk3qYlTVmNRdcNnZuKiBXwy1/9voRjgMSKsyQlF3b4spJkFAUM5KWerEkHOExGpBuf0K5DFApJM8zu9J4YkfCaiGBM7fv/EBVJOA08z2d5yMcuG/2XdWPmXTkJDHisSYo3ozI8ZVBHMaoF9KghWbKoNwoLqRSEeIoGw0uVlDdiL9y6b1plpW6Z9d16pX+VdFMEROAZVYIMLUAc3oAGaAIMn8AJm4M14Nl6NmfH+gxaM/M8h+CPj4xsTr5id</latexit>

Sε ⇣

<latexit sha1_base64="/UzE65jcPU3hD4VqaVWabqpFoc=">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</latexit><latexit sha1_base64="/UzE65jcPU3hD4VqaVWabqpFoc=">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</latexit><latexit sha1_base64="/UzE65jcPU3hD4VqaVWabqpFoc=">ADVHicpZLPb9MwFMeTlsEosHVw5GJRIQ0NUkRYhekMThwQkOjW6W2ihznJX2aYxvbqVpF/dv4L5C4wxX+AQ4XaSxHxIH3un5fb/W+hrJ4qjsWH4zW+1b23cvrN5t3Pv/oOt7e7OwxMjS81gyCSXepRQAxwFDC1aDiOlgRYJh9Pk7G2tn85BG5Tik10qmBY0F5gho9aN4h1/NEkgR1HB53I9er7qHMeTOdWgDHIpJoeY7zaenNoZaEhXnQkKxsUck3VDJkZW40FeU1ehGTwkoQk3COMo9ojM8B8Zp0S9fdZMa2CoXE0QSI1muCdliqRi0DaWNDgmGMKJsgwN0FOS2OQihgLZWIp+LKv0sztBZFeYJAabnIC2pLOvxH/F0XI+MbcxNHjfHsfHYRY9zthf1wXeR6EzVNz2vqKO5+n6SlQUIyzg1ZhyFyk4rqi0yDm5naUBRdkZzGKdzVEbQAsy0Wqx/wYo8zaQmjoisz3/bK1oYsywS5ykcs7mq1cObtHFps/1phUKVFgRzFqdlJSdWkvozkRQ1MuXrqFMowMlbEY1ZdYle2lL8wqLOpboagjXm5NBPwr70cdB7+CwCWjTe+w98Xa9yHvlHXjvSNv6DH/i/D/+n/an1t/W632xvn1pbf3HnkXar21h9qzRK</latexit>

Compute Sinkhorn batch loss: X(imp) X(imp) ηrX(imp)Sε ⇣

<latexit sha1_base64="z+sl9kaENViUnkBLas0AGLRkKhY=">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</latexit><latexit sha1_base64="z+sl9kaENViUnkBLas0AGLRkKhY=">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</latexit><latexit sha1_base64="z+sl9kaENViUnkBLas0AGLRkKhY=">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</latexit>

Update imputations:

ˆ X = (1 M) X(obs) + M X(imp)

<latexit sha1_base64="fm0/34Hnzc/TWR2SYZVdXMz3S7Q=">ACRXicbZBLS8NAFIUnPmt9V26GSxCi1gSEXQjFN24ESrYBzS1TKaTdugkE2ZuiXk54lL1/4GcVfdaloj1ta7OnPON3DvcQLBNZjmi7GwuLS8spZy65vbG5t53Z2a1qGirIqlUKqhkM0E9xnVeAgWCNQjHiOYHWnfzXO6wOmNJf+HQwD1vJI1+cupwQSq51r2z0Cke0R6Dlu1IhjfIELFj7GP9ZNXLRlR8Ivch8VpKOLMT6agvA8xL2gGLdzebNkTgbPCysVeZROpZ17tjuSh7zgQqidMyA2hFRAGngsVZO9QsILRPuqzZGfBA+8RjuhU9TLqI8aErFYew5P3NB4RT+uh5yTMeE09m43N/7JmCO5K+J+EALzaYIkmRsKDBKPK8UdrhgFMUwEoYoni2LaI4pQSIrPJg1Ys/fOi9pJyTJL1u1pvnyZdpFB+gAFZCFzlAZXaMKqiKHtErGqF348l4M0bGxze6YKR/9tCfMT6/AI4Nsl0=</latexit><latexit sha1_base64="fm0/34Hnzc/TWR2SYZVdXMz3S7Q=">ACRXicbZBLS8NAFIUnPmt9V26GSxCi1gSEXQjFN24ESrYBzS1TKaTdugkE2ZuiXk54lL1/4GcVfdaloj1ta7OnPON3DvcQLBNZjmi7GwuLS8spZy65vbG5t53Z2a1qGirIqlUKqhkM0E9xnVeAgWCNQjHiOYHWnfzXO6wOmNJf+HQwD1vJI1+cupwQSq51r2z0Cke0R6Dlu1IhjfIELFj7GP9ZNXLRlR8Ivch8VpKOLMT6agvA8xL2gGLdzebNkTgbPCysVeZROpZ17tjuSh7zgQqidMyA2hFRAGngsVZO9QsILRPuqzZGfBA+8RjuhU9TLqI8aErFYew5P3NB4RT+uh5yTMeE09m43N/7JmCO5K+J+EALzaYIkmRsKDBKPK8UdrhgFMUwEoYoni2LaI4pQSIrPJg1Ys/fOi9pJyTJL1u1pvnyZdpFB+gAFZCFzlAZXaMKqiKHtErGqF348l4M0bGxze6YKR/9tCfMT6/AI4Nsl0=</latexit><latexit sha1_base64="fm0/34Hnzc/TWR2SYZVdXMz3S7Q=">ACRXicbZBLS8NAFIUnPmt9V26GSxCi1gSEXQjFN24ESrYBzS1TKaTdugkE2ZuiXk54lL1/4GcVfdaloj1ta7OnPON3DvcQLBNZjmi7GwuLS8spZy65vbG5t53Z2a1qGirIqlUKqhkM0E9xnVeAgWCNQjHiOYHWnfzXO6wOmNJf+HQwD1vJI1+cupwQSq51r2z0Cke0R6Dlu1IhjfIELFj7GP9ZNXLRlR8Ivch8VpKOLMT6agvA8xL2gGLdzebNkTgbPCysVeZROpZ17tjuSh7zgQqidMyA2hFRAGngsVZO9QsILRPuqzZGfBA+8RjuhU9TLqI8aErFYew5P3NB4RT+uh5yTMeE09m43N/7JmCO5K+J+EALzaYIkmRsKDBKPK8UdrhgFMUwEoYoni2LaI4pQSIrPJg1Ys/fOi9pJyTJL1u1pvnyZdpFB+gAFZCFzlAZXaMKqiKHtErGqF348l4M0bGxze6YKR/9tCfMT6/AI4Nsl0=</latexit>
  • Output:

(mij = 1 ⇐ ⇒ xij missing)

<latexit sha1_base64="heLOp3lOm1h7mqdxFLRb6WFJWc=">ACDnicbVDLTgJBEJzF+IL9ehlIjHBC9k1JnoxIXrxiIk8EiBkduiFkZndzUwvgWz2F4wfY7yhV2+e/RuXx0HBOlVXVSfd5YZSGLTtbyuztr6xuZXdzu3s7u0f5A+PaiaINIcqD2SgGy4zIUPVRQoRFqYMqVUHcHd1O/PgRtROA/4jiEtmI9X3iCM0ylTr5YVJ1YPCX0hjq0JTyPjuZzC2GEMVXCGOH3kvNOvmCX7BnoKnEWpEAWqHTyX61uwCMFPnLJjGk6dojtmGkUXEKSa0UGQsYHrAfN7lCExmcKTDsezZ5K6JkXaIp9oLP5dzxmypixctOMYtg3y95U/M9rRuhdt2PhxGCz9NI6nmRpBjQaTe0KzRwlOUMK5FeijlfaYZx7TBXNqAs/zvKqldlBy75DxcFsq3iy6y5ISckiJxyBUpk3tSIVXCyTN5JRPybr1Yb9bE+phHM9Zi5j8gfX5AyAYm48=</latexit><latexit sha1_base64="heLOp3lOm1h7mqdxFLRb6WFJWc=">ACDnicbVDLTgJBEJzF+IL9ehlIjHBC9k1JnoxIXrxiIk8EiBkduiFkZndzUwvgWz2F4wfY7yhV2+e/RuXx0HBOlVXVSfd5YZSGLTtbyuztr6xuZXdzu3s7u0f5A+PaiaINIcqD2SgGy4zIUPVRQoRFqYMqVUHcHd1O/PgRtROA/4jiEtmI9X3iCM0ylTr5YVJ1YPCX0hjq0JTyPjuZzC2GEMVXCGOH3kvNOvmCX7BnoKnEWpEAWqHTyX61uwCMFPnLJjGk6dojtmGkUXEKSa0UGQsYHrAfN7lCExmcKTDsezZ5K6JkXaIp9oLP5dzxmypixctOMYtg3y95U/M9rRuhdt2PhxGCz9NI6nmRpBjQaTe0KzRwlOUMK5FeijlfaYZx7TBXNqAs/zvKqldlBy75DxcFsq3iy6y5ISckiJxyBUpk3tSIVXCyTN5JRPybr1Yb9bE+phHM9Zi5j8gfX5AyAYm48=</latexit><latexit sha1_base64="heLOp3lOm1h7mqdxFLRb6WFJWc=">ACDnicbVDLTgJBEJzF+IL9ehlIjHBC9k1JnoxIXrxiIk8EiBkduiFkZndzUwvgWz2F4wfY7yhV2+e/RuXx0HBOlVXVSfd5YZSGLTtbyuztr6xuZXdzu3s7u0f5A+PaiaINIcqD2SgGy4zIUPVRQoRFqYMqVUHcHd1O/PgRtROA/4jiEtmI9X3iCM0ylTr5YVJ1YPCX0hjq0JTyPjuZzC2GEMVXCGOH3kvNOvmCX7BnoKnEWpEAWqHTyX61uwCMFPnLJjGk6dojtmGkUXEKSa0UGQsYHrAfN7lCExmcKTDsezZ5K6JkXaIp9oLP5dzxmypixctOMYtg3y95U/M9rRuhdt2PhxGCz9NI6nmRpBjQaTe0KzRwlOUMK5FeijlfaYZx7TBXNqAs/zvKqldlBy75DxcFsq3iy6y5ISckiJxyBUpk3tSIVXCyTN5JRPybr1Yb9bE+phHM9Zi5j8gfX5AyAYm48=</latexit>

X = (1 M) X(obs) + M NA

<latexit sha1_base64="fcLQsmpahu58Dhkw0FtSEC852U8=">ACN3icbZBNS8NAEIY3ftb6VfXoZbEIFbEkIuhFqHrxoihYLbS1bLaTdukmG3YnpSXkd4m/wrMH8aZe/QemtaJW5/TOvM/AzOuGUhi07UdrYnJqemY2M5edX1hcWs6trF4bFWkOZa6k0hWXGZAigDIKlFAJNTDflXDjdk4G/k0XtBEquMJ+CHWftQLhCc4wHTVylzWfYdv14kpCD2nB2fnqz5ItWlNhfSbuI0LyjVbCd2m39gXhdD+PwoaeTydtEeFv0rnJHIk1FdNHIPtabikQ8BcsmMqTp2iPWYaRcQpKtRQZCxjusBdVmV4QmYD6Yetwbfp/QTU9pim2gw/4nHjPfmL7vpszgXDPuDYb/edUIvYN6LIwQgh4iqSeF0mKig5CpE2hgaPsp4JxLdJDKW8zTimUWfTBJzxf/+K692iYxedy7186XiURYaskw1SIA7ZJyVySi5ImXByR57IC3m17q1n68V6+0QnrNHOGvlV1vsH3y2r/Q=</latexit><latexit sha1_base64="fcLQsmpahu58Dhkw0FtSEC852U8=">ACN3icbZBNS8NAEIY3ftb6VfXoZbEIFbEkIuhFqHrxoihYLbS1bLaTdukmG3YnpSXkd4m/wrMH8aZe/QemtaJW5/TOvM/AzOuGUhi07UdrYnJqemY2M5edX1hcWs6trF4bFWkOZa6k0hWXGZAigDIKlFAJNTDflXDjdk4G/k0XtBEquMJ+CHWftQLhCc4wHTVylzWfYdv14kpCD2nB2fnqz5ItWlNhfSbuI0LyjVbCd2m39gXhdD+PwoaeTydtEeFv0rnJHIk1FdNHIPtabikQ8BcsmMqTp2iPWYaRcQpKtRQZCxjusBdVmV4QmYD6Yetwbfp/QTU9pim2gw/4nHjPfmL7vpszgXDPuDYb/edUIvYN6LIwQgh4iqSeF0mKig5CpE2hgaPsp4JxLdJDKW8zTimUWfTBJzxf/+K692iYxedy7186XiURYaskw1SIA7ZJyVySi5ImXByR57IC3m17q1n68V6+0QnrNHOGvlV1vsH3y2r/Q=</latexit><latexit sha1_base64="fcLQsmpahu58Dhkw0FtSEC852U8=">ACN3icbZBNS8NAEIY3ftb6VfXoZbEIFbEkIuhFqHrxoihYLbS1bLaTdukmG3YnpSXkd4m/wrMH8aZe/QemtaJW5/TOvM/AzOuGUhi07UdrYnJqemY2M5edX1hcWs6trF4bFWkOZa6k0hWXGZAigDIKlFAJNTDflXDjdk4G/k0XtBEquMJ+CHWftQLhCc4wHTVylzWfYdv14kpCD2nB2fnqz5ItWlNhfSbuI0LyjVbCd2m39gXhdD+PwoaeTydtEeFv0rnJHIk1FdNHIPtabikQ8BcsmMqTp2iPWYaRcQpKtRQZCxjusBdVmV4QmYD6Yetwbfp/QTU9pim2gw/4nHjPfmL7vpszgXDPuDYb/edUIvYN6LIwQgh4iqSeF0mKig5CpE2hgaPsp4JxLdJDKW8zTimUWfTBJzxf/+K692iYxedy7186XiURYaskw1SIA7ZJyVySi5ImXByR57IC3m17q1n68V6+0QnrNHOGvlV1vsH3y2r/Q=</latexit>

M ∈ {0, 1}n×d

<latexit sha1_base64="/8sny1EYQ4IX4hWD2QYdkiBHlCU=">ACXicbVDLSsNAFJ34rPVdelmsAgupCQi6Loxo1QwT6giWUyuWmHTiZhZlIsIR8gfoy4q279ANf+jZPahbae1bn3nMu9/gJZ0rb9pe1tLyurZe2ihvbm3v7Fb29lsqTiWFJo15LDs+UcCZgKZmkMnkUAin0PbH14XensEUrFY3OtxAl5E+oKFjBJtWr1K1Y2IHvhdptjlwnsZvYpdtz8IROuZhEoHOTGZdfsKfAicWakimZo9CqfbhDTNAKhKSdKdR070V5GpGaUQ152UwUJoUPSh24wYokSxGzysfpPzk+DmOJ9QDwtP5tz0ik1Djyjae4W81rRfM/rZvq8NLmEhSDYIai9HClGMd4yIWHDAJVPOxIYRKZg7FdEAkodqEVzYJOP/LpLWc2xa87debV+NcuihA7RETpBDrpAdXSDGqiJKHpCL2iC3qxn69WaWO8/1iVrNnOA/sD6+Aajdpmq</latexit><latexit sha1_base64="/8sny1EYQ4IX4hWD2QYdkiBHlCU=">ACXicbVDLSsNAFJ34rPVdelmsAgupCQi6Loxo1QwT6giWUyuWmHTiZhZlIsIR8gfoy4q279ANf+jZPahbae1bn3nMu9/gJZ0rb9pe1tLyurZe2ihvbm3v7Fb29lsqTiWFJo15LDs+UcCZgKZmkMnkUAin0PbH14XensEUrFY3OtxAl5E+oKFjBJtWr1K1Y2IHvhdptjlwnsZvYpdtz8IROuZhEoHOTGZdfsKfAicWakimZo9CqfbhDTNAKhKSdKdR070V5GpGaUQ152UwUJoUPSh24wYokSxGzysfpPzk+DmOJ9QDwtP5tz0ik1Djyjae4W81rRfM/rZvq8NLmEhSDYIai9HClGMd4yIWHDAJVPOxIYRKZg7FdEAkodqEVzYJOP/LpLWc2xa87debV+NcuihA7RETpBDrpAdXSDGqiJKHpCL2iC3qxn69WaWO8/1iVrNnOA/sD6+Aajdpmq</latexit><latexit sha1_base64="/8sny1EYQ4IX4hWD2QYdkiBHlCU=">ACXicbVDLSsNAFJ34rPVdelmsAgupCQi6Loxo1QwT6giWUyuWmHTiZhZlIsIR8gfoy4q279ANf+jZPahbae1bn3nMu9/gJZ0rb9pe1tLyurZe2ihvbm3v7Fb29lsqTiWFJo15LDs+UcCZgKZmkMnkUAin0PbH14XensEUrFY3OtxAl5E+oKFjBJtWr1K1Y2IHvhdptjlwnsZvYpdtz8IROuZhEoHOTGZdfsKfAicWakimZo9CqfbhDTNAKhKSdKdR070V5GpGaUQ152UwUJoUPSh24wYokSxGzysfpPzk+DmOJ9QDwtP5tz0ik1Djyjae4W81rRfM/rZvq8NLmEhSDYIai9HClGMd4yIWHDAJVPOxIYRKZg7FdEAkodqEVzYJOP/LpLWc2xa87debV+NcuihA7RETpBDrpAdXSDGqiJKHpCL2iC3qxn69WaWO8/1iVrNnOA/sD6+Aajdpmq</latexit>

,

slide-12
SLIDE 12

Observed values vs dimension

  • But 2 sampled batches should still have similar distributions.

D ⇣

' 0

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  • Problem: as dimension increases, almost all entries have NAs.

Example with 25% missing rate: 2d 3d 6d 10d

slide-13
SLIDE 13

Imputation Algorithm

X = (1 M) X(obs) + M NA

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  • Input:
  • Initial imputations:

(column mean of observed values + noise)

x(imp)

ij

= x(obs)

:j

+ ε if mij = 1

<latexit sha1_base64="275QwYfKOwL8E6YFBoi50yO7uO4=">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</latexit><latexit sha1_base64="275QwYfKOwL8E6YFBoi50yO7uO4=">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</latexit><latexit sha1_base64="275QwYfKOwL8E6YFBoi50yO7uO4=">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</latexit>

for t = 1, 2..., T

<latexit sha1_base64="QLAB3J5VfvCcCAbYNkuD24RIVHw=">ACAnicbVDLSgNBEJz1GeNr1aMeBoPgISy7QdCLEPTiMUJekCxhdtKbDJl9MNMbEpZcxI8Rb9Gr/+DZv3ETc9DEOlV3VUN1ebEUGm37y1hb39jc2s7t5Hf39g8OzaPjuo4SxaHGIxmpsc0SBFCDQVKaMYKWOBJaHiD+5neGILSIgqrOI7BDVgvFL7gDLNVxzxrI4zQ81M/UnRCkd5Sp0hLlmUVabVjFmzLnoOuEmdBCmSBSsf8bHcjngQIpdM65Zjx+imTKHgEib5dqIhZnzAetDqDkWsQxaAdtPR/JEJvZilwD7Q+fzbnrJA63HgZ6AYV8va7Plf1orQf/GTUYJwghzyZ5ieSYkRnfdCuUMBRjPCuBJZUMr7TDGOWv5rAFn+d9VUi9Zjm05j1eF8t2ixw5JefkjkmpTJA6mQGuHkibyQKXkzno1XY2q8/1jXjMXNCfkD4+MblL6U7Q=</latexit><latexit sha1_base64="QLAB3J5VfvCcCAbYNkuD24RIVHw=">ACAnicbVDLSgNBEJz1GeNr1aMeBoPgISy7QdCLEPTiMUJekCxhdtKbDJl9MNMbEpZcxI8Rb9Gr/+DZv3ETc9DEOlV3VUN1ebEUGm37y1hb39jc2s7t5Hf39g8OzaPjuo4SxaHGIxmpsc0SBFCDQVKaMYKWOBJaHiD+5neGILSIgqrOI7BDVgvFL7gDLNVxzxrI4zQ81M/UnRCkd5Sp0hLlmUVabVjFmzLnoOuEmdBCmSBSsf8bHcjngQIpdM65Zjx+imTKHgEib5dqIhZnzAetDqDkWsQxaAdtPR/JEJvZilwD7Q+fzbnrJA63HgZ6AYV8va7Plf1orQf/GTUYJwghzyZ5ieSYkRnfdCuUMBRjPCuBJZUMr7TDGOWv5rAFn+d9VUi9Zjm05j1eF8t2ixw5JefkjkmpTJA6mQGuHkibyQKXkzno1XY2q8/1jXjMXNCfkD4+MblL6U7Q=</latexit><latexit sha1_base64="QLAB3J5VfvCcCAbYNkuD24RIVHw=">ACAnicbVDLSgNBEJz1GeNr1aMeBoPgISy7QdCLEPTiMUJekCxhdtKbDJl9MNMbEpZcxI8Rb9Gr/+DZv3ETc9DEOlV3VUN1ebEUGm37y1hb39jc2s7t5Hf39g8OzaPjuo4SxaHGIxmpsc0SBFCDQVKaMYKWOBJaHiD+5neGILSIgqrOI7BDVgvFL7gDLNVxzxrI4zQ81M/UnRCkd5Sp0hLlmUVabVjFmzLnoOuEmdBCmSBSsf8bHcjngQIpdM65Zjx+imTKHgEib5dqIhZnzAetDqDkWsQxaAdtPR/JEJvZilwD7Q+fzbnrJA63HgZ6AYV8va7Plf1orQf/GTUYJwghzyZ5ieSYkRnfdCuUMBRjPCuBJZUMr7TDGOWv5rAFn+d9VUi9Zjm05j1eF8t2ixw5JefkjkmpTJA6mQGuHkibyQKXkzno1XY2q8/1jXjMXNCfkD4+MblL6U7Q=</latexit>
  • ˆ

X = (1 M) X(obs) + M X(imp)

<latexit sha1_base64="fm0/34Hnzc/TWR2SYZVdXMz3S7Q=">ACRXicbZBLS8NAFIUnPmt9V26GSxCi1gSEXQjFN24ESrYBzS1TKaTdugkE2ZuiXk54lL1/4GcVfdaloj1ta7OnPON3DvcQLBNZjmi7GwuLS8spZy65vbG5t53Z2a1qGirIqlUKqhkM0E9xnVeAgWCNQjHiOYHWnfzXO6wOmNJf+HQwD1vJI1+cupwQSq51r2z0Cke0R6Dlu1IhjfIELFj7GP9ZNXLRlR8Ivch8VpKOLMT6agvA8xL2gGLdzebNkTgbPCysVeZROpZ17tjuSh7zgQqidMyA2hFRAGngsVZO9QsILRPuqzZGfBA+8RjuhU9TLqI8aErFYew5P3NB4RT+uh5yTMeE09m43N/7JmCO5K+J+EALzaYIkmRsKDBKPK8UdrhgFMUwEoYoni2LaI4pQSIrPJg1Ys/fOi9pJyTJL1u1pvnyZdpFB+gAFZCFzlAZXaMKqiKHtErGqF348l4M0bGxze6YKR/9tCfMT6/AI4Nsl0=</latexit><latexit sha1_base64="fm0/34Hnzc/TWR2SYZVdXMz3S7Q=">ACRXicbZBLS8NAFIUnPmt9V26GSxCi1gSEXQjFN24ESrYBzS1TKaTdugkE2ZuiXk54lL1/4GcVfdaloj1ta7OnPON3DvcQLBNZjmi7GwuLS8spZy65vbG5t53Z2a1qGirIqlUKqhkM0E9xnVeAgWCNQjHiOYHWnfzXO6wOmNJf+HQwD1vJI1+cupwQSq51r2z0Cke0R6Dlu1IhjfIELFj7GP9ZNXLRlR8Ivch8VpKOLMT6agvA8xL2gGLdzebNkTgbPCysVeZROpZ17tjuSh7zgQqidMyA2hFRAGngsVZO9QsILRPuqzZGfBA+8RjuhU9TLqI8aErFYew5P3NB4RT+uh5yTMeE09m43N/7JmCO5K+J+EALzaYIkmRsKDBKPK8UdrhgFMUwEoYoni2LaI4pQSIrPJg1Ys/fOi9pJyTJL1u1pvnyZdpFB+gAFZCFzlAZXaMKqiKHtErGqF348l4M0bGxze6YKR/9tCfMT6/AI4Nsl0=</latexit><latexit sha1_base64="fm0/34Hnzc/TWR2SYZVdXMz3S7Q=">ACRXicbZBLS8NAFIUnPmt9V26GSxCi1gSEXQjFN24ESrYBzS1TKaTdugkE2ZuiXk54lL1/4GcVfdaloj1ta7OnPON3DvcQLBNZjmi7GwuLS8spZy65vbG5t53Z2a1qGirIqlUKqhkM0E9xnVeAgWCNQjHiOYHWnfzXO6wOmNJf+HQwD1vJI1+cupwQSq51r2z0Cke0R6Dlu1IhjfIELFj7GP9ZNXLRlR8Ivch8VpKOLMT6agvA8xL2gGLdzebNkTgbPCysVeZROpZ17tjuSh7zgQqidMyA2hFRAGngsVZO9QsILRPuqzZGfBA+8RjuhU9TLqI8aErFYew5P3NB4RT+uh5yTMeE09m43N/7JmCO5K+J+EALzaYIkmRsKDBKPK8UdrhgFMUwEoYoni2LaI4pQSIrPJg1Ys/fOi9pJyTJL1u1pvnyZdpFB+gAFZCFzlAZXaMKqiKHtErGqF348l4M0bGxze6YKR/9tCfMT6/AI4Nsl0=</latexit>
  • Output:

Compute Sinkhorn batch loss:

Sε ⇣

<latexit sha1_base64="uxG5FbYQr7oNWCXC3hI250OJZb4=">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</latexit><latexit sha1_base64="uxG5FbYQr7oNWCXC3hI250OJZb4=">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</latexit><latexit sha1_base64="uxG5FbYQr7oNWCXC3hI250OJZb4=">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</latexit>

Update imputations:

X(imp) X(imp) ηrX(imp)Sε ⇣

<latexit sha1_base64="eytFb9t+VsPeN+S3K1DWLBrpzqU=">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</latexit><latexit sha1_base64="eytFb9t+VsPeN+S3K1DWLBrpzqU=">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</latexit><latexit sha1_base64="eytFb9t+VsPeN+S3K1DWLBrpzqU=">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</latexit>

Sample 2 batches:

ˆ Xi1,...,iK

<latexit sha1_base64="0M51O1OBUIsRWjEkbXcKz9Mwgw=">ACBnicbVDLSsNAFJ34rPUVdelmsAguSkhE0GXRjeCmgn1AE8JkOmGTh7M3BRLyFb8GHFX3foHrv0bk5qFtp7VufecC+ceLxFcgWl+aSura+sbm7Wt+vbO7t6+fnDYVXEqKevQWMSy7xHFBI9YBzgI1k8kI6EnWM8b35R6b8Kk4nH0ANOEOSEZRdznlECxcnVsBwQyOyQeH7Wz3M3467VxIZhNDF373JXb5iGOQdeJlZFGqhC29U/7WFM05BFQAVRamCZCTgZkcCpYHndThVLCB2TERsMJzxREQmZcrLH+TM5PvVjiSFgeD7/tmckVGoaeoWnzKsWtXL5nzZIwb9yMh4lKbCIFpZC81OBIcZlJ3jIJaMgpgUhVPIiKYBkYRC0Vy9aMBa/HeZdM8NyzSs+4tG67rqoaO0Qk6Qxa6RC10i9qogyh6Qi9oht60Z+1Vm2nvP9YVrbo5Qn+gfXwD6b+YFw=</latexit><latexit sha1_base64="0M51O1OBUIsRWjEkbXcKz9Mwgw=">ACBnicbVDLSsNAFJ34rPUVdelmsAguSkhE0GXRjeCmgn1AE8JkOmGTh7M3BRLyFb8GHFX3foHrv0bk5qFtp7VufecC+ceLxFcgWl+aSura+sbm7Wt+vbO7t6+fnDYVXEqKevQWMSy7xHFBI9YBzgI1k8kI6EnWM8b35R6b8Kk4nH0ANOEOSEZRdznlECxcnVsBwQyOyQeH7Wz3M3467VxIZhNDF373JXb5iGOQdeJlZFGqhC29U/7WFM05BFQAVRamCZCTgZkcCpYHndThVLCB2TERsMJzxREQmZcrLH+TM5PvVjiSFgeD7/tmckVGoaeoWnzKsWtXL5nzZIwb9yMh4lKbCIFpZC81OBIcZlJ3jIJaMgpgUhVPIiKYBkYRC0Vy9aMBa/HeZdM8NyzSs+4tG67rqoaO0Qk6Qxa6RC10i9qogyh6Qi9oht60Z+1Vm2nvP9YVrbo5Qn+gfXwD6b+YFw=</latexit><latexit sha1_base64="0M51O1OBUIsRWjEkbXcKz9Mwgw=">ACBnicbVDLSsNAFJ34rPUVdelmsAguSkhE0GXRjeCmgn1AE8JkOmGTh7M3BRLyFb8GHFX3foHrv0bk5qFtp7VufecC+ceLxFcgWl+aSura+sbm7Wt+vbO7t6+fnDYVXEqKevQWMSy7xHFBI9YBzgI1k8kI6EnWM8b35R6b8Kk4nH0ANOEOSEZRdznlECxcnVsBwQyOyQeH7Wz3M3467VxIZhNDF373JXb5iGOQdeJlZFGqhC29U/7WFM05BFQAVRamCZCTgZkcCpYHndThVLCB2TERsMJzxREQmZcrLH+TM5PvVjiSFgeD7/tmckVGoaeoWnzKsWtXL5nzZIwb9yMh4lKbCIFpZC81OBIcZlJ3jIJaMgpgUhVPIiKYBkYRC0Vy9aMBa/HeZdM8NyzSs+4tG67rqoaO0Qk6Qxa6RC10i9qogyh6Qi9oht60Z+1Vm2nvP9YVrbo5Qn+gfXwD6b+YFw=</latexit>

ˆ Xj1,...,jK

<latexit sha1_base64="Ub2/RDrFRpUkuS9+KYEb3LP1DQ=">ACBnicbVDLSsNAFL3xWesr6tLNYBFclJCIoMuiG8FNBfuAJoTJdNJO3kwMymWkK34MeKuvUPXPs3JjULbT2rc+85F849XsyZVKb5pa2srq1vbFa2qts7u3v7+sFhW0aJILRFIh6Jrocl5SykLcUp91YUBx4nHa8U2hdyZUSBaFD2oaUyfAg5D5jGCVr1wd2UOsUjvAauj5aTfL3HTkWnVkGEYdjdy7zNVrpmHOgZaJVZIalGi6+qfdj0gS0FARjqXsWasnBQLxQinWdVOJI0xGeMB7fUnLJYhDqh0sf5Mxk69SOB1JCi+fzbnuJAymng5Z4ir1zUiuV/Wi9R/pWTsjBOFA1Jbsk1P+FIRajoBPWZoETxaU4wESwPisgQC0xU3lw1b8Ba/HeZtM8NyzSs+4ta47rsogLHcAJnYMElNOAWmtACAk/wAjN40561V2mvf9YV7Ty5gj+QPv4BuzRmBk=</latexit><latexit sha1_base64="Ub2/RDrFRpUkuS9+KYEb3LP1DQ=">ACBnicbVDLSsNAFL3xWesr6tLNYBFclJCIoMuiG8FNBfuAJoTJdNJO3kwMymWkK34MeKuvUPXPs3JjULbT2rc+85F849XsyZVKb5pa2srq1vbFa2qts7u3v7+sFhW0aJILRFIh6Jrocl5SykLcUp91YUBx4nHa8U2hdyZUSBaFD2oaUyfAg5D5jGCVr1wd2UOsUjvAauj5aTfL3HTkWnVkGEYdjdy7zNVrpmHOgZaJVZIalGi6+qfdj0gS0FARjqXsWasnBQLxQinWdVOJI0xGeMB7fUnLJYhDqh0sf5Mxk69SOB1JCi+fzbnuJAymng5Z4ir1zUiuV/Wi9R/pWTsjBOFA1Jbsk1P+FIRajoBPWZoETxaU4wESwPisgQC0xU3lw1b8Ba/HeZtM8NyzSs+4ta47rsogLHcAJnYMElNOAWmtACAk/wAjN40561V2mvf9YV7Ty5gj+QPv4BuzRmBk=</latexit><latexit sha1_base64="Ub2/RDrFRpUkuS9+KYEb3LP1DQ=">ACBnicbVDLSsNAFL3xWesr6tLNYBFclJCIoMuiG8FNBfuAJoTJdNJO3kwMymWkK34MeKuvUPXPs3JjULbT2rc+85F849XsyZVKb5pa2srq1vbFa2qts7u3v7+sFhW0aJILRFIh6Jrocl5SykLcUp91YUBx4nHa8U2hdyZUSBaFD2oaUyfAg5D5jGCVr1wd2UOsUjvAauj5aTfL3HTkWnVkGEYdjdy7zNVrpmHOgZaJVZIalGi6+qfdj0gS0FARjqXsWasnBQLxQinWdVOJI0xGeMB7fUnLJYhDqh0sf5Mxk69SOB1JCi+fzbnuJAymng5Z4ir1zUiuV/Wi9R/pWTsjBOFA1Jbsk1P+FIRajoBPWZoETxaU4wESwPisgQC0xU3lw1b8Ba/HeZtM8NyzSs+4ta47rsogLHcAJnYMElNOAWmtACAk/wAjN40561V2mvf9YV7Ty5gj+QPv4BuzRmBk=</latexit>

and

ˆ X (1 M) X(obs) + M X(imp)

<latexit sha1_base64="Pud5PcJeo2rU13x5Jbjw+4Ajo=">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</latexit><latexit sha1_base64="Pud5PcJeo2rU13x5Jbjw+4Ajo=">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</latexit><latexit sha1_base64="Pud5PcJeo2rU13x5Jbjw+4Ajo=">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</latexit>

Mix observations and imputations:

M ∈ {0, 1}n×d

<latexit sha1_base64="/8sny1EYQ4IX4hWD2QYdkiBHlCU=">ACXicbVDLSsNAFJ34rPVdelmsAgupCQi6Loxo1QwT6giWUyuWmHTiZhZlIsIR8gfoy4q279ANf+jZPahbae1bn3nMu9/gJZ0rb9pe1tLyurZe2ihvbm3v7Fb29lsqTiWFJo15LDs+UcCZgKZmkMnkUAin0PbH14XensEUrFY3OtxAl5E+oKFjBJtWr1K1Y2IHvhdptjlwnsZvYpdtz8IROuZhEoHOTGZdfsKfAicWakimZo9CqfbhDTNAKhKSdKdR070V5GpGaUQ152UwUJoUPSh24wYokSxGzysfpPzk+DmOJ9QDwtP5tz0ik1Djyjae4W81rRfM/rZvq8NLmEhSDYIai9HClGMd4yIWHDAJVPOxIYRKZg7FdEAkodqEVzYJOP/LpLWc2xa87debV+NcuihA7RETpBDrpAdXSDGqiJKHpCL2iC3qxn69WaWO8/1iVrNnOA/sD6+Aajdpmq</latexit><latexit sha1_base64="/8sny1EYQ4IX4hWD2QYdkiBHlCU=">ACXicbVDLSsNAFJ34rPVdelmsAgupCQi6Loxo1QwT6giWUyuWmHTiZhZlIsIR8gfoy4q279ANf+jZPahbae1bn3nMu9/gJZ0rb9pe1tLyurZe2ihvbm3v7Fb29lsqTiWFJo15LDs+UcCZgKZmkMnkUAin0PbH14XensEUrFY3OtxAl5E+oKFjBJtWr1K1Y2IHvhdptjlwnsZvYpdtz8IROuZhEoHOTGZdfsKfAicWakimZo9CqfbhDTNAKhKSdKdR070V5GpGaUQ152UwUJoUPSh24wYokSxGzysfpPzk+DmOJ9QDwtP5tz0ik1Djyjae4W81rRfM/rZvq8NLmEhSDYIai9HClGMd4yIWHDAJVPOxIYRKZg7FdEAkodqEVzYJOP/LpLWc2xa87debV+NcuihA7RETpBDrpAdXSDGqiJKHpCL2iC3qxn69WaWO8/1iVrNnOA/sD6+Aajdpmq</latexit><latexit sha1_base64="/8sny1EYQ4IX4hWD2QYdkiBHlCU=">ACXicbVDLSsNAFJ34rPVdelmsAgupCQi6Loxo1QwT6giWUyuWmHTiZhZlIsIR8gfoy4q279ANf+jZPahbae1bn3nMu9/gJZ0rb9pe1tLyurZe2ihvbm3v7Fb29lsqTiWFJo15LDs+UcCZgKZmkMnkUAin0PbH14XensEUrFY3OtxAl5E+oKFjBJtWr1K1Y2IHvhdptjlwnsZvYpdtz8IROuZhEoHOTGZdfsKfAicWakimZo9CqfbhDTNAKhKSdKdR070V5GpGaUQ152UwUJoUPSh24wYokSxGzysfpPzk+DmOJ9QDwtP5tz0ik1Djyjae4W81rRfM/rZvq8NLmEhSDYIai9HClGMd4yIWHDAJVPOxIYRKZg7FdEAkodqEVzYJOP/LpLWc2xa87debV+NcuihA7RETpBDrpAdXSDGqiJKHpCL2iC3qxn69WaWO8/1iVrNnOA/sD6+Aajdpmq</latexit>

,

(mij = 1 ⇐ ⇒ xij missing)

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slide-14
SLIDE 14

What if we want a parametric model?

slide-15
SLIDE 15

OT as an imputation criterion

Sε ⇣

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Sinkhorn batch loss

Imputer( · , θ)

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+M

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(1 M)

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X(imp)

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X(obs)

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X(imp)

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for t = 1, 2..., T

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Batch ˆ Xi1,...,iK

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Batch ˆ Xj1,...,jK

<latexit sha1_base64="r7RWL956I8s9PnUzSPMGd/E7jQk=">ACFHicbVDLSsNAFJ34rPVdelmsAguSkhE0KXUjeBGwWqhCWEyvWmnTyYuSmWkL8QP0bcqVtd+zemNQtfZ3XuPefCucdPpdBoWR/G3PzC4tJybaW+ura+sdnY2r7RSaY4dHgiE9X1mQYpYuigQAndVAGLfAm3/uhsqt+OQWmRxNc4ScGN2CAWgeAMy5XMB2EO8zbDHlIC+qEDHMnYhj6Qd4tCi8fenaLmqbZokPvovAaTcu0ZqB/iV2RJqlw6TXenX7Cswhi5Jp3bOtFN2cKRcQlF3Mg0p4yM2gF5/LFIdswi0m9/NXivofpAoiHQ2fzdnrNI60nkl5pXv1bmy7/03oZBiduLuI0Q4h5aSm1IJMUEzptiPaFAo5yUhLGlSiDUh4yxTiWPdbLBuzf/4lN4embZn21VHztF1USO7ZI8cEJsck1NyTi5Jh3ByTx7JM3kxHown49l4/bLOGdXNDvkB4+0ToZOd4A=</latexit><latexit sha1_base64="r7RWL956I8s9PnUzSPMGd/E7jQk=">ACFHicbVDLSsNAFJ34rPVdelmsAguSkhE0KXUjeBGwWqhCWEyvWmnTyYuSmWkL8QP0bcqVtd+zemNQtfZ3XuPefCucdPpdBoWR/G3PzC4tJybaW+ura+sdnY2r7RSaY4dHgiE9X1mQYpYuigQAndVAGLfAm3/uhsqt+OQWmRxNc4ScGN2CAWgeAMy5XMB2EO8zbDHlIC+qEDHMnYhj6Qd4tCi8fenaLmqbZokPvovAaTcu0ZqB/iV2RJqlw6TXenX7Cswhi5Jp3bOtFN2cKRcQlF3Mg0p4yM2gF5/LFIdswi0m9/NXivofpAoiHQ2fzdnrNI60nkl5pXv1bmy7/03oZBiduLuI0Q4h5aSm1IJMUEzptiPaFAo5yUhLGlSiDUh4yxTiWPdbLBuzf/4lN4embZn21VHztF1USO7ZI8cEJsck1NyTi5Jh3ByTx7JM3kxHown49l4/bLOGdXNDvkB4+0ToZOd4A=</latexit><latexit sha1_base64="r7RWL956I8s9PnUzSPMGd/E7jQk=">ACFHicbVDLSsNAFJ34rPVdelmsAguSkhE0KXUjeBGwWqhCWEyvWmnTyYuSmWkL8QP0bcqVtd+zemNQtfZ3XuPefCucdPpdBoWR/G3PzC4tJybaW+ura+sdnY2r7RSaY4dHgiE9X1mQYpYuigQAndVAGLfAm3/uhsqt+OQWmRxNc4ScGN2CAWgeAMy5XMB2EO8zbDHlIC+qEDHMnYhj6Qd4tCi8fenaLmqbZokPvovAaTcu0ZqB/iV2RJqlw6TXenX7Cswhi5Jp3bOtFN2cKRcQlF3Mg0p4yM2gF5/LFIdswi0m9/NXivofpAoiHQ2fzdnrNI60nkl5pXv1bmy7/03oZBiduLuI0Q4h5aSm1IJMUEzptiPaFAo5yUhLGlSiDUh4yxTiWPdbLBuzf/4lN4embZn21VHztF1USO7ZI8cEJsck1NyTi5Jh3ByTx7JM3kxHown49l4/bLOGdXNDvkB4+0ToZOd4A=</latexit>

θ θ ηrθSε

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  • We used OT to directly fit imputation values by gradient descent.

We could use it to fit any parametric imputation model.

e.g. linear model, MLP , …

Allows out-of-sample imputation

slide-16
SLIDE 16

OT as an imputation criterion

Sε ⇣

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Sinkhorn batch loss

Imputer( · , θ)

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+M

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(1 M)

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X(imp)

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X(obs)

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X(imp)

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for t = 1, 2..., T

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Batch ˆ Xi1,...,iK

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Batch ˆ Xj1,...,jK

<latexit sha1_base64="r7RWL956I8s9PnUzSPMGd/E7jQk=">ACFHicbVDLSsNAFJ34rPVdelmsAguSkhE0KXUjeBGwWqhCWEyvWmnTyYuSmWkL8QP0bcqVtd+zemNQtfZ3XuPefCucdPpdBoWR/G3PzC4tJybaW+ura+sdnY2r7RSaY4dHgiE9X1mQYpYuigQAndVAGLfAm3/uhsqt+OQWmRxNc4ScGN2CAWgeAMy5XMB2EO8zbDHlIC+qEDHMnYhj6Qd4tCi8fenaLmqbZokPvovAaTcu0ZqB/iV2RJqlw6TXenX7Cswhi5Jp3bOtFN2cKRcQlF3Mg0p4yM2gF5/LFIdswi0m9/NXivofpAoiHQ2fzdnrNI60nkl5pXv1bmy7/03oZBiduLuI0Q4h5aSm1IJMUEzptiPaFAo5yUhLGlSiDUh4yxTiWPdbLBuzf/4lN4embZn21VHztF1USO7ZI8cEJsck1NyTi5Jh3ByTx7JM3kxHown49l4/bLOGdXNDvkB4+0ToZOd4A=</latexit><latexit sha1_base64="r7RWL956I8s9PnUzSPMGd/E7jQk=">ACFHicbVDLSsNAFJ34rPVdelmsAguSkhE0KXUjeBGwWqhCWEyvWmnTyYuSmWkL8QP0bcqVtd+zemNQtfZ3XuPefCucdPpdBoWR/G3PzC4tJybaW+ura+sdnY2r7RSaY4dHgiE9X1mQYpYuigQAndVAGLfAm3/uhsqt+OQWmRxNc4ScGN2CAWgeAMy5XMB2EO8zbDHlIC+qEDHMnYhj6Qd4tCi8fenaLmqbZokPvovAaTcu0ZqB/iV2RJqlw6TXenX7Cswhi5Jp3bOtFN2cKRcQlF3Mg0p4yM2gF5/LFIdswi0m9/NXivofpAoiHQ2fzdnrNI60nkl5pXv1bmy7/03oZBiduLuI0Q4h5aSm1IJMUEzptiPaFAo5yUhLGlSiDUh4yxTiWPdbLBuzf/4lN4embZn21VHztF1USO7ZI8cEJsck1NyTi5Jh3ByTx7JM3kxHown49l4/bLOGdXNDvkB4+0ToZOd4A=</latexit><latexit sha1_base64="r7RWL956I8s9PnUzSPMGd/E7jQk=">ACFHicbVDLSsNAFJ34rPVdelmsAguSkhE0KXUjeBGwWqhCWEyvWmnTyYuSmWkL8QP0bcqVtd+zemNQtfZ3XuPefCucdPpdBoWR/G3PzC4tJybaW+ura+sdnY2r7RSaY4dHgiE9X1mQYpYuigQAndVAGLfAm3/uhsqt+OQWmRxNc4ScGN2CAWgeAMy5XMB2EO8zbDHlIC+qEDHMnYhj6Qd4tCi8fenaLmqbZokPvovAaTcu0ZqB/iV2RJqlw6TXenX7Cswhi5Jp3bOtFN2cKRcQlF3Mg0p4yM2gF5/LFIdswi0m9/NXivofpAoiHQ2fzdnrNI60nkl5pXv1bmy7/03oZBiduLuI0Q4h5aSm1IJMUEzptiPaFAo5yUhLGlSiDUh4yxTiWPdbLBuzf/4lN4embZn21VHztF1USO7ZI8cEJsck1NyTi5Jh3ByTx7JM3kxHown49l4/bLOGdXNDvkB4+0ToZOd4A=</latexit>

θ θ ηrθSε

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How to implement in practice?

  • We used OT to directly fit imputation values by gradient descent.

We could use it to fit any parametric imputation model.

e.g. linear model, MLP , …

Allows out-of-sample imputation

slide-17
SLIDE 17

Round-robin imputation

Imputer( · , θj)

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XT

:j

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XT

:d

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XT

:1

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XT

:2

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X(imp)

:j

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Imputer( · , θj+1)

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XT

:d

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XT

:1

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XT

:2

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(1 M) XT (obs)

:j

+ M XT (imp)

:j

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X(imp)

:j+1

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X(imp)

:j−1

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Sε ⇣

<latexit sha1_base64="uxG5FbYQr7oNWCXC3hI250OJZb4=">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</latexit><latexit sha1_base64="uxG5FbYQr7oNWCXC3hI250OJZb4=">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</latexit><latexit sha1_base64="uxG5FbYQr7oNWCXC3hI250OJZb4=">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</latexit>

Sinkhorn batch loss θj θj ηrθjSε

<latexit sha1_base64="XVITvbC9D719srnDN0odnykyAIU=">ACKXicbVDLSgNBEJz1bXxFPXoZDIXw64IehS9eFQ0GsiGpXfSm4yZfTDTGw1LPkj8GPGmXv0RJzGKGutU1XTdFeYKWnIdV+dqemZ2bn5hcXS0vLK6lp5fePapLkWBOpSnU9BINKJlgjSQrmUaIQ4U3Yfd06N/0UBuZJlfUz7AZQzuRkRAVgrKpz51kC45b7CiEDr9I5/a3vct4T7CYQKguJLH/DLwO+BxsxINZxScavuCHySeGNSYWOcB+Unv5WKPMaEhAJjGp6bUbMATVIoHJT83GAGogtbLR6MjMJxGiaxf3o3gHfiVLN7S589P4ZLyA2ph+HNhMDdcxfbyj+5zVyio6ahUynDARNmK9KFecUj6sjbekRkGqbwkILe2iXHRAgyBbsk24P29d5Jc71c9t+pdHFSOT8ZdLAts12mcO2TE7Y+esxgR7YE/shb06j86z8+K8fUanPGfTfYLzvsH4bSnqQ=</latexit><latexit sha1_base64="XVITvbC9D719srnDN0odnykyAIU=">ACKXicbVDLSgNBEJz1bXxFPXoZDIXw64IehS9eFQ0GsiGpXfSm4yZfTDTGw1LPkj8GPGmXv0RJzGKGutU1XTdFeYKWnIdV+dqemZ2bn5hcXS0vLK6lp5fePapLkWBOpSnU9BINKJlgjSQrmUaIQ4U3Yfd06N/0UBuZJlfUz7AZQzuRkRAVgrKpz51kC45b7CiEDr9I5/a3vct4T7CYQKguJLH/DLwO+BxsxINZxScavuCHySeGNSYWOcB+Unv5WKPMaEhAJjGp6bUbMATVIoHJT83GAGogtbLR6MjMJxGiaxf3o3gHfiVLN7S589P4ZLyA2ph+HNhMDdcxfbyj+5zVyio6ahUynDARNmK9KFecUj6sjbekRkGqbwkILe2iXHRAgyBbsk24P29d5Jc71c9t+pdHFSOT8ZdLAts12mcO2TE7Y+esxgR7YE/shb06j86z8+K8fUanPGfTfYLzvsH4bSnqQ=</latexit><latexit sha1_base64="XVITvbC9D719srnDN0odnykyAIU=">ACKXicbVDLSgNBEJz1bXxFPXoZDIXw64IehS9eFQ0GsiGpXfSm4yZfTDTGw1LPkj8GPGmXv0RJzGKGutU1XTdFeYKWnIdV+dqemZ2bn5hcXS0vLK6lp5fePapLkWBOpSnU9BINKJlgjSQrmUaIQ4U3Yfd06N/0UBuZJlfUz7AZQzuRkRAVgrKpz51kC45b7CiEDr9I5/a3vct4T7CYQKguJLH/DLwO+BxsxINZxScavuCHySeGNSYWOcB+Unv5WKPMaEhAJjGp6bUbMATVIoHJT83GAGogtbLR6MjMJxGiaxf3o3gHfiVLN7S589P4ZLyA2ph+HNhMDdcxfbyj+5zVyio6ahUynDARNmK9KFecUj6sjbekRkGqbwkILe2iXHRAgyBbsk24P29d5Jc71c9t+pdHFSOT8ZdLAts12mcO2TE7Y+esxgR7YE/shb06j86z8+K8fUanPGfTfYLzvsH4bSnqQ=</latexit>

for i = 1, ..., R

<latexit sha1_base64="CJNKhXxXL4J3x5b4uoAHtjSzUzw=">ACAXicbVDLTgJBEJzF+IL9chlIjHxQDa7xkQvJkQvHtHI4ENmR16YcLsIzO9BLhYPwY4w29+hGe/RsX3IOCdaruqk6qy42k0GhZX0ZubX1jcyu/XdjZ3ds/KB4eNXQYKw51HspQtVymQYoA6ihQitSwHxXQtMd3s715giUFmHwiJMIHJ/1A+EJzjBdYulDsIYXS/xQkWnVNBraleoaZoV+tAtli3TWoCuEjsjZKh1i1+dnohj30IkEumdu2InQSplBwCdNCJ9YQMT5kfWj3RiLSAfNBO8l48ceUns5D4ADoYv5tT5iv9cR3U4/PcKCXtfnyP60do3flJCKIYoSAp5ZU82JMaTzOmhPKOAoJylhXIk0KOUDphjHtLRC2oC9/O8qaZybtmXa9xfl6k3WRZ6UyAk5Iza5JFVyR2qkTjh5Ii9kRt6MZ+PVmBnvP9ackd0ckz8wPr4BCEWUpA=</latexit><latexit sha1_base64="CJNKhXxXL4J3x5b4uoAHtjSzUzw=">ACAXicbVDLTgJBEJzF+IL9chlIjHxQDa7xkQvJkQvHtHI4ENmR16YcLsIzO9BLhYPwY4w29+hGe/RsX3IOCdaruqk6qy42k0GhZX0ZubX1jcyu/XdjZ3ds/KB4eNXQYKw51HspQtVymQYoA6ihQitSwHxXQtMd3s715giUFmHwiJMIHJ/1A+EJzjBdYulDsIYXS/xQkWnVNBraleoaZoV+tAtli3TWoCuEjsjZKh1i1+dnohj30IkEumdu2InQSplBwCdNCJ9YQMT5kfWj3RiLSAfNBO8l48ceUns5D4ADoYv5tT5iv9cR3U4/PcKCXtfnyP60do3flJCKIYoSAp5ZU82JMaTzOmhPKOAoJylhXIk0KOUDphjHtLRC2oC9/O8qaZybtmXa9xfl6k3WRZ6UyAk5Iza5JFVyR2qkTjh5Ii9kRt6MZ+PVmBnvP9ackd0ckz8wPr4BCEWUpA=</latexit><latexit sha1_base64="CJNKhXxXL4J3x5b4uoAHtjSzUzw=">ACAXicbVDLTgJBEJzF+IL9chlIjHxQDa7xkQvJkQvHtHI4ENmR16YcLsIzO9BLhYPwY4w29+hGe/RsX3IOCdaruqk6qy42k0GhZX0ZubX1jcyu/XdjZ3ds/KB4eNXQYKw51HspQtVymQYoA6ihQitSwHxXQtMd3s715giUFmHwiJMIHJ/1A+EJzjBdYulDsIYXS/xQkWnVNBraleoaZoV+tAtli3TWoCuEjsjZKh1i1+dnohj30IkEumdu2InQSplBwCdNCJ9YQMT5kfWj3RiLSAfNBO8l48ceUns5D4ADoYv5tT5iv9cR3U4/PcKCXtfnyP60do3flJCKIYoSAp5ZU82JMaTzOmhPKOAoJylhXIk0KOUDphjHtLRC2oC9/O8qaZybtmXa9xfl6k3WRZ6UyAk5Iza5JFVyR2qkTjh5Ii9kRt6MZ+PVmBnvP9ackd0ckz8wPr4BCEWUpA=</latexit>

Batch ˆ Xi1,...,iK

<latexit sha1_base64="XVbo7TI6XErkPxDVB3nugsW8kjA=">ACFHicbVDLSsNAFJ34tr6iLt0MFsFCYkIuhTdCG4q2Ac0JUymN2Zw8mDmprSE/IX4MeJO3eravzGtWjrWZ17z7lw7vFTKTa9pexsLi0vLK6tl7b2Nza3jF39o6yRSHFk9koro+0yBFDC0UKGbKmCRL6HjP1xN9M4QlBZJfIfjFPoRu49FIDjDcuWZloswvySIQ9pQd2QYe5GDEM/yLtF4eXCcxrUsqwGFd5N4Zl127KnoPEqUidVGh65qc7SHgWQYxcMq17jp1iP2cKBZdQ1NxMQ8r4A7uH3mAoUh2zCHQ/H01fK+hRkCiKIdDp/Nues0jrceSXnklePatNlv9pvQyD834u4jRDiHlpKbUgkxQTOmIDoQCjnJcEsaVKINSHjLFOJY91soGnNl/50n7xHJsy7k9rV9cVl2skQNySI6JQ87IBbkmTdIinDySZ/JK3own48V4Nd5/rAtGdbNP/sD4+AaegZ3e</latexit><latexit sha1_base64="XVbo7TI6XErkPxDVB3nugsW8kjA=">ACFHicbVDLSsNAFJ34tr6iLt0MFsFCYkIuhTdCG4q2Ac0JUymN2Zw8mDmprSE/IX4MeJO3eravzGtWjrWZ17z7lw7vFTKTa9pexsLi0vLK6tl7b2Nza3jF39o6yRSHFk9koro+0yBFDC0UKGbKmCRL6HjP1xN9M4QlBZJfIfjFPoRu49FIDjDcuWZloswvySIQ9pQd2QYe5GDEM/yLtF4eXCcxrUsqwGFd5N4Zl127KnoPEqUidVGh65qc7SHgWQYxcMq17jp1iP2cKBZdQ1NxMQ8r4A7uH3mAoUh2zCHQ/H01fK+hRkCiKIdDp/Nues0jrceSXnklePatNlv9pvQyD834u4jRDiHlpKbUgkxQTOmIDoQCjnJcEsaVKINSHjLFOJY91soGnNl/50n7xHJsy7k9rV9cVl2skQNySI6JQ87IBbkmTdIinDySZ/JK3own48V4Nd5/rAtGdbNP/sD4+AaegZ3e</latexit><latexit sha1_base64="XVbo7TI6XErkPxDVB3nugsW8kjA=">ACFHicbVDLSsNAFJ34tr6iLt0MFsFCYkIuhTdCG4q2Ac0JUymN2Zw8mDmprSE/IX4MeJO3eravzGtWjrWZ17z7lw7vFTKTa9pexsLi0vLK6tl7b2Nza3jF39o6yRSHFk9koro+0yBFDC0UKGbKmCRL6HjP1xN9M4QlBZJfIfjFPoRu49FIDjDcuWZloswvySIQ9pQd2QYe5GDEM/yLtF4eXCcxrUsqwGFd5N4Zl127KnoPEqUidVGh65qc7SHgWQYxcMq17jp1iP2cKBZdQ1NxMQ8r4A7uH3mAoUh2zCHQ/H01fK+hRkCiKIdDp/Nues0jrceSXnklePatNlv9pvQyD834u4jRDiHlpKbUgkxQTOmIDoQCjnJcEsaVKINSHjLFOJY91soGnNl/50n7xHJsy7k9rV9cVl2skQNySI6JQ87IBbkmTdIinDySZ/JK3own48V4Nd5/rAtGdbNP/sD4+AaegZ3e</latexit>

Batch ˆ Xj1,...,jK

<latexit sha1_base64="r7RWL956I8s9PnUzSPMGd/E7jQk=">ACFHicbVDLSsNAFJ34rPVdelmsAguSkhE0KXUjeBGwWqhCWEyvWmnTyYuSmWkL8QP0bcqVtd+zemNQtfZ3XuPefCucdPpdBoWR/G3PzC4tJybaW+ura+sdnY2r7RSaY4dHgiE9X1mQYpYuigQAndVAGLfAm3/uhsqt+OQWmRxNc4ScGN2CAWgeAMy5XMB2EO8zbDHlIC+qEDHMnYhj6Qd4tCi8fenaLmqbZokPvovAaTcu0ZqB/iV2RJqlw6TXenX7Cswhi5Jp3bOtFN2cKRcQlF3Mg0p4yM2gF5/LFIdswi0m9/NXivofpAoiHQ2fzdnrNI60nkl5pXv1bmy7/03oZBiduLuI0Q4h5aSm1IJMUEzptiPaFAo5yUhLGlSiDUh4yxTiWPdbLBuzf/4lN4embZn21VHztF1USO7ZI8cEJsck1NyTi5Jh3ByTx7JM3kxHown49l4/bLOGdXNDvkB4+0ToZOd4A=</latexit><latexit sha1_base64="r7RWL956I8s9PnUzSPMGd/E7jQk=">ACFHicbVDLSsNAFJ34rPVdelmsAguSkhE0KXUjeBGwWqhCWEyvWmnTyYuSmWkL8QP0bcqVtd+zemNQtfZ3XuPefCucdPpdBoWR/G3PzC4tJybaW+ura+sdnY2r7RSaY4dHgiE9X1mQYpYuigQAndVAGLfAm3/uhsqt+OQWmRxNc4ScGN2CAWgeAMy5XMB2EO8zbDHlIC+qEDHMnYhj6Qd4tCi8fenaLmqbZokPvovAaTcu0ZqB/iV2RJqlw6TXenX7Cswhi5Jp3bOtFN2cKRcQlF3Mg0p4yM2gF5/LFIdswi0m9/NXivofpAoiHQ2fzdnrNI60nkl5pXv1bmy7/03oZBiduLuI0Q4h5aSm1IJMUEzptiPaFAo5yUhLGlSiDUh4yxTiWPdbLBuzf/4lN4embZn21VHztF1USO7ZI8cEJsck1NyTi5Jh3ByTx7JM3kxHown49l4/bLOGdXNDvkB4+0ToZOd4A=</latexit><latexit sha1_base64="r7RWL956I8s9PnUzSPMGd/E7jQk=">ACFHicbVDLSsNAFJ34rPVdelmsAguSkhE0KXUjeBGwWqhCWEyvWmnTyYuSmWkL8QP0bcqVtd+zemNQtfZ3XuPefCucdPpdBoWR/G3PzC4tJybaW+ura+sdnY2r7RSaY4dHgiE9X1mQYpYuigQAndVAGLfAm3/uhsqt+OQWmRxNc4ScGN2CAWgeAMy5XMB2EO8zbDHlIC+qEDHMnYhj6Qd4tCi8fenaLmqbZokPvovAaTcu0ZqB/iV2RJqlw6TXenX7Cswhi5Jp3bOtFN2cKRcQlF3Mg0p4yM2gF5/LFIdswi0m9/NXivofpAoiHQ2fzdnrNI60nkl5pXv1bmy7/03oZBiduLuI0Q4h5aSm1IJMUEzptiPaFAo5yUhLGlSiDUh4yxTiWPdbLBuzf/4lN4embZn21VHztF1USO7ZI8cEJsck1NyTi5Jh3ByTx7JM3kxHown49l4/bLOGdXNDvkB4+0ToZOd4A=</latexit>

Impute variables one by one, using all other variables as inputs

θ1, θ2, ..., θd

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Use d

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parametric models (one for each variable) Generalization of Imputation by Chained Equations (e.g. R’s mice)

slide-18
SLIDE 18

Comparison with baselines

Extensive experiments on UCI datasets in MCAR, MAR and MNAR settings. Three performance metrics:

  • Mean Absolute Error (MAE)
  • Root Mean Square Error (RMSE)
  • Optimal Transport ( )

W 2

2

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50% MCAR:

slide-19
SLIDE 19

Comparison with Deep Learning

30% MNAR:

(masked quantiles)

MIWAE (Mattei & Frellsen, 2019), GAIN (Yoon et al., 2018), VAEs (Ivanov et al., 2019 )

github.com/BorisMuzellec/MissingDataOT