Rémi Gribonval Inria Rennes - Bretagne Atlantique · R. GRIBONVAL - CSA 2015 - Berlin Example: Compressive Clustering 9 X Compressive Gaussian Mixture Estimation Anthony Bourrier12,Remi

Rémi Gribonval Inria Rennes - Bretagne Atlantique

[email protected]

mailto:[email protected]

R. GRIBONVAL - CSA 2015 - Berlin

Contributors & Collaborators

2

Anthony Bourrier Nicolas Keriven Yann Traonmilin

Tomer Peleg

Gilles Puy

Mike Davies Patrick PerezGilles Blanchard


Agenda

From Compressive Sensing to Compressive Learning ? Information-preserving projections & sketches Compressive Clustering / Compressive GMM Conclusion

3


Machine Learning

Available data training collection of feature vectors = point cloud

Goals infer parameters to achieve a certain task generalization to future samples with the same probability distribution

Examples

4

Compressive Gaussian Mixture Estimation

Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez

11 Technicolor, 975 Avenue des Champs Blancs, 35576 Cesson Sevigne, France

[email protected]

2 INRIA Rennes - Bretagne Atlantique, Campus de Beaulieu, 35042 Rennes, [email protected]

Motivation

Goal: Infer parameters ✓ from n-dimensional data X = {x1

, . . . ,xN}. Thistypically requires extensive access to the data. Proposed method: Inferfrom a sketch of the data ) memory and privacy savings.

n x

1

. . .xN

Learning set(size Nn)

ˆ

A

=) mˆ

z

Database sketch(size m)

L=) K ✓

Learned parameters(size K)

Figure 1: Illustration of the proposed sketching framework. A is a sketch-

ing operator, L is a learning method from the sketch.

Model and problem statement

Application to mixture of isotropic Gaussians in Rn:

fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)

Data X = {xj}Nj=1 ⇠i.i.d.

p =

Pks=1↵sfµs

with:

•weights ↵1

, . . . ,↵k (positive, sum to one)•means µ

1

, . . . ,µk 2 Rn.

Sketch = Fourier samplings at different frequencies: (Af )l = ˆf (!l).Empirical version: ( ˆA(X ))l =

1

N

PNj=1 exp(�ih!l,xji) ⇡ (Ap)l.

We want to infer the mixture parameters from ˆ

z =

ˆ

A(X ).Problem casted as:

p = argminq2⌃k

kˆz�Aqk22

, (2)

where ⌃k = mixtures of k isotropic Gaussians with positive weights.Standard CS Our problem

Signal x 2 Rn f 2 L1

(Rn)

Dimension n InfiniteSparsity k k

Dictionary {e1

, . . . , en} F = {fµ,µ 2 Rn}Measurements x 7! ha,xi f 7!

RRn f (x)e�ih!,xidx

Algorithm

Current estimate p with weights {↵s}ks=1 and support ˆ� = {ˆµs}ks=1.Residual ˆr = ˆ

z�Ap.1. Searching new support functions:

Search for ”good components to add” to the support) Local minima of µ 7! �hAfµ, ˆri, added to the support ˆ�.New support ˆ�0.

2. k-term thresholding:Projection of ˆz onto ˆ

�

0 with positivity constraints on coefficients:

argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1

, . . . , ˆµK].k highest coefficients and corresponding support are kept! new support ˆ� and coefficients ↵

1

, . . . , ↵k.3. Final ”shift”:

Gradient descent algorithm on the objective function, with initialization atthe current support and coefficients.

First step Second step Third step

Figure 2: Algorithm illustration in dimension n = 1 for k = 3 Gaus-

sians. Top: Iteration 1. Bottom: Iteration 2. Blue curve=true mixture,

Red curve=reconstructed mixture, Green curve=gradient function. Green

Dots=Candidate Centroids, Red Dots=Reconstructed Centroids.

Experimental results

Data setup: � = 1, (↵1

, . . . ,↵k) drawn uniformly on the simplex.Entries of µ

1

, . . . ,µk ⇠i.i.d.

N (0, 1).

Algorithm heuristics:•Frequencies drawn i.i.d. from N (0, Id).

•New support function search (step 1) initialized as ru, where r uniformly

drawn in0,max

x2X||x||

2

�and u uniformly drawn on B

2

(0, 1).

Comparison between:•Our method: Sketch is computed on-the-fly and data is discarded.

•EM: Data is stored to allow the standard optimization steps to be per-formed.

Quality measures: KL Divergence and Hellinger distance.

NCompressed EM

KL div. Hell. Mem. KL div. Hell. Mem.10

3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24

Table 1: Comparison between our method and an EM algorithm. n =

20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size m

k*n/

m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Figure 3: Left: Example of data and sketch for n = 2. Right: Reconstruc-

tion quality for n = 10.

-6 -4 -2 0 2 4 6-4

-3

-2

-1

0

1

2

3

PCA principal subspace

Dictionary learning dictionary

Clustering centroids

Classification classifier parameters

(e.g. support vectors)

X


Point cloud = large matrix of feature vectors

Challenging dimensions

5

X




5

x1X




5

x1 x2X




5

x1 x2 xN…X X



High feature dimension n Large collection size N


5

x1 x2 xN…X X



High feature dimension n Large collection size N


5

x1 x2 xN…X X

Challenge: compress before learning ?X


Compressive Machine Learning ?


6

x1 x2 xN…X X

yNy2 …y1Y = MX

M




Reduce feature dimension [Calderbank & al 2009, Reboredo & al 2013]

(Random) feature projection Exploits / needs low-dimensional feature model

6

x1 x2 xN…X X

yNy2 …y1Y = MX


Challenges of large collections

Feature projection: limited impact

7

X

Y = MX


Challenges of large collections

Feature projection: limited impact

7

X

Y = MX

“Big Data” Challenge: compress collection size



Point cloud = … empirical probability distribution

8

X




Reduce collection dimension coresets

see e.g. [Agarwal & al 2003, Felman 2010]

sketching & hashing see e.g. [Thaper & al 2002, Cormode & al 2005]

8

X







8

XM

z 2 Rm

Sketching operator nonlinear in the feature vectors

linear in their probability distribution







8

XM

z 2 Rm

Sketching operator nonlinear in the feature vectors



Example: Compressive Clustering

9

X


Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez


[email protected]


Motivation



n x

1

. . .xN


ˆ

A

=) mˆ

z


L=) K ✓






fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)


p =

Pks=1↵sfµs

with:

•weights ↵1


1

, . . . ,µk 2 Rn.


1

N



z =

ˆ


p = argminq2⌃k

kˆz�Aqk22

, (2)



(Rn)


Dictionary {e1



Algorithm





�


argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1


1

, . . . , ↵k.3. Final ”shift”:










1

, . . . ,µk ⇠i.i.d.

N (0, 1).



drawn in0,max

x2X||x||

2


2

(0, 1).




NCompressed EM


3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24


20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size m

k*n/

m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16



M z 2 Rm

Recovery algorithm

estimated centroids

ground truth

N = 1000;n = 2 m = 60


Computational impact of sketching

10

Ph.D. A. Bourrier & N. Keriven

Computation time Memory

Collection size N Collection size N

Tim

e (s

)

Mem

ory

(byt

es)


Data distribution

Sketch

The Sketch Trick

11

X ⇠ p(x)

z` =1

N

NX

i=1

h`(xi)


Data distribution

Sketch

The Sketch Trick

11

X ⇠ p(x)

z` =1

N

NX

i=1

h`(xi)

⇡ Eh`(X)


Data distribution

Sketch

The Sketch Trick

11

X ⇠ p(x)

z` =1

N

NX

i=1

h`(xi)

⇡ Eh`(X)

=

Zh`(x)p(x)dx


Data distribution

Sketch

The Sketch Trick

11

X ⇠ p(x)

z` =1

N

NX

i=1

h`(xi)

⇡ Eh`(X)

=

Zh`(x)p(x)dx

nonlinear in the feature vectors linear in the distribution p(x)


Data distribution

Sketch

The Sketch Trick

11

X ⇠ p(x)

z` =1

N

NX

i=1

h`(xi)

⇡ Eh`(X)

=

Zh`(x)p(x)dx

y

Signal

space

x

Observation space

Signal Processing inverse problems compressive sensing

MM

Probability space

Sketch space

Machine Learning method of moments compressive learning

z

p

Linear “projection”



Information preservation ?

Data distribution

Sketch

The Sketch Trick

11

X ⇠ p(x)

z` =1

N

NX

i=1

h`(xi)

⇡ Eh`(X)

=

Zh`(x)p(x)dx

y

Signal

space

x

Observation space


MM

Probability space

Sketch space


z

p




The Sketch Trick

Data distribution

Sketch

Dimension reduction ?

12

X ⇠ p(x)

z` =1

N

NX

i=1

h`(xi)

⇡ Eh`(X)

=

Zh`(x)p(x)dx

y

Signal

space

x

Observation space


MM

Probability space

Sketch space


z

p



Information preserving projections


Stable recovery

Signal space

Observation space


x

Ex: set of k-sparse vectors

14

M

y

m ⌧ nRm

Rn

Model set = signals of interest

⌃

⌃k = {x 2 Rn, kxk0 k}


Stable recovery

Signal space

Observation space


x


14

M

y

m ⌧ nRm

Rn


⌃

⌃k = {x 2 Rn, kxk0 k}

Recovery algorithm

= “decoder”

�

Ideal goal: build decoder with the guarantee that

(instance optimality [Cohen & al 2009])

�

kx��(Mx+ e)k Ckek, 8x 2 ⌃


Stable recovery

Signal space

Observation space


x


14

M

y

m ⌧ nRm

Rn


⌃

⌃k = {x 2 Rn, kxk0 k}

Recovery algorithm

= “decoder”

�



�


Are there such decoders?


Stable recovery of k-sparse vectors

Typical decoders L1 minimization

LASSO [Tibshirani 1994],Basis Pursuit [Chen & al 1999]

Greedy algorithms (Orthonormal) Matching Pursuit [Mallat & Zhang 1993], Iterative Hard Thresholding (IHT) [Blumensath & Davies 2009], …

Guarantees Assume Restricted isometry property

[Candès & al 2004]

Exact recovery Stability to noise Robustness to model error

15

�(y) := argminx:Mx=y

kxk1

1� � kMzk22kzk22

1 + �

when kzk0 2k


Stable recovery

Low-dimensional model Sparse

16

Signal space

Observation space


x

M

y

m ⌧ nRm

Rn


⌃


Stable recovery

Low-dimensional model Sparse Sparse in dictionary D

Signal space

Observation space


x

M

y

m ⌧ nRm

Rn


⌃

17


Stable recovery

Low-dimensional model Sparse Sparse in dictionary D Co-sparse in analysis operator A

total variation,

physics-driven sparse models ..

18

Signal space

Observation space


x

M

y

m ⌧ nRm

Rn


⌃


Stable recovery


total variation,

physics-driven sparse models …

Low-rank matrix or tensor matrix completion,

phase-retrieval,

blind sensor calibration …

19

Signal space

Observation space


x

M

y

m ⌧ nRm

Rn


⌃



total variation,



phase-retrieval,


Manifold / Union of manifolds detection, estimation,

localization, mapping …

Matrix with sparse inverse Gaussian graphical models

Given point cloud database indexing

Stable recovery

20

Signal space

Observation space


x

M

y

m ⌧ nRm

Rn


⌃



total variation,



phase-retrieval,


Manifold / Union of manifolds detection, estimation,

localization, mapping …

Matrix with sparse inverse Gaussian graphical models

Given point cloud database indexing

Gaussian Mixture Model…

Stable recovery

21

Signal space

Observation space


x

M

y

m ⌧ nRm

Rn


⌃

Vector spaceH


Low-dimensional model

arbitrary set

General stable recovery

22

Observation space


x

M

y

m ⌧ nRm


⌃

⌃ ⇢ H

Signal space RnVector spaceH

Recovery algorithm

= “decoder”

�



�


Are there such decoders?


Theorem 1: RIP is necessary Definition: (general) Restricted Isometry Property (RIP) on secant set

RIP holds as soon as there exists an instance optimal decoder

Stable recovery from arbitrary model sets

23

↵ kMzkkzk � when z 2 ⌃� ⌃ := {x� x

0, x, x

0 2 ⌃}up to renormalization

↵ =p1� �;� =

p1 + �




Theorem 2: RIP is sufficient RIP implies existence of decoder with performance guarantees:

Exact recovery

Stable to noise

Bonus: robust to model error

[Cohen & al 2009] for

[Bourrier & al 2014] for arbitrary

kx��(Mx+ e)k C(�)kek+ C

0(�)d⌃(x,⌃)


23


0, x, x


↵ =p1� �;� =

p1 + �

⌃k

⌃





Exact recovery

Stable to noise





0(�)d⌃(x,⌃)


23


0, x, x


↵ =p1� �;� =

p1 + �

⌃k

⌃





Exact recovery

Stable to noise





0(�)d⌃(x,⌃)


23


0, x, x


↵ =p1� �;� =

p1 + �

⌃k

⌃ Distance to model set

Compressive Learning Examples


Compressive Machine Learning

Point cloud = empirical probability distribution

Reduce collection dimension = sketching

25

X

z` =1

N

NX

i=1

h`(xi) 1 ` m

M z 2 Rm

Sketching operator

Choosing information preserving sketch ?


Goal: find k centroids

Standard approach = K-means

Sketching approach

p(x) is spatially localized

need “incoherent” sampling choose Fourier sampling

sample characteristic function

choose sampling frequencies


26


Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez


[email protected]


Motivation



n x

1

. . .xN


ˆ

A

=) mˆ

z


L=) K ✓






fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)


p =

Pks=1↵sfµs

with:

•weights ↵1


1

, . . . ,µk 2 Rn.


1

N



z =

ˆ


p = argminq2⌃k

kˆz�Aqk22

, (2)



(Rn)


Dictionary {e1



Algorithm





�


argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1


1

, . . . , ↵k.3. Final ”shift”:










1

, . . . ,µk ⇠i.i.d.

N (0, 1).



drawn in0,max

x2X||x||

2


2

(0, 1).




NCompressed EM


3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24


20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size mk*

n/m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16



z`

=1

N

NX

i=1

ejw>` xi

!` 2 Rn




Sketching approach






26


Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez


[email protected]


Motivation



n x

1

. . .xN


ˆ

A

=) mˆ

z


L=) K ✓






fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)


p =

Pks=1↵sfµs

with:

•weights ↵1


1

, . . . ,µk 2 Rn.


1

N



z =

ˆ


p = argminq2⌃k

kˆz�Aqk22

, (2)



(Rn)


Dictionary {e1



Algorithm





�


argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1


1

, . . . , ↵k.3. Final ”shift”:










1

, . . . ,µk ⇠i.i.d.

N (0, 1).



drawn in0,max

x2X||x||

2


2

(0, 1).




NCompressed EM


3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24


20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size mk*

n/m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16



z`

=1

N

NX

i=1

ejw>` xi

!` 2 Rn




Sketching approach






26


Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez


[email protected]


Motivation



n x

1

. . .xN


ˆ

A

=) mˆ

z


L=) K ✓






fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)


p =

Pks=1↵sfµs

with:

•weights ↵1


1

, . . . ,µk 2 Rn.


1

N



z =

ˆ


p = argminq2⌃k

kˆz�Aqk22

, (2)



(Rn)


Dictionary {e1



Algorithm





�


argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1


1

, . . . , ↵k.3. Final ”shift”:










1

, . . . ,µk ⇠i.i.d.

N (0, 1).



drawn in0,max

x2X||x||

2


2

(0, 1).




NCompressed EM


3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24


20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size mk*

n/m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16



z`

=1

N

NX

i=1

ejw>` xi

!` 2 Rn




Sketching approach






26


Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez


[email protected]


Motivation



n x

1

. . .xN


ˆ

A

=) mˆ

z


L=) K ✓






fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)


p =

Pks=1↵sfµs

with:

•weights ↵1


1

, . . . ,µk 2 Rn.


1

N



z =

ˆ


p = argminq2⌃k

kˆz�Aqk22

, (2)



(Rn)


Dictionary {e1



Algorithm





�


argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1


1

, . . . , ↵k.3. Final ”shift”:










1

, . . . ,µk ⇠i.i.d.

N (0, 1).



drawn in0,max

x2X||x||

2


2

(0, 1).




NCompressed EM


3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24


20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size mk*

n/m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16



z`

=1

N

NX

i=1

ejw>` xi

!` 2 Rn

How ? see poster N. Keriven



27

X M z 2 Rm

N = 1000;n = 2Sampled

Characteristic Function m = 60


Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez


[email protected]


Motivation



n x

1

. . .xN


ˆ

A

=) mˆ

z


L=) K ✓






fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)


p =

Pks=1↵sfµs

with:

•weights ↵1


1

, . . . ,µk 2 Rn.


1

N



z =

ˆ


p = argminq2⌃k

kˆz�Aqk22

, (2)



(Rn)


Dictionary {e1



Algorithm





�


argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1


1

, . . . , ↵k.3. Final ”shift”:










1

, . . . ,µk ⇠i.i.d.

N (0, 1).



drawn in0,max

x2X||x||

2


2

(0, 1).




NCompressed EM


3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24


20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size mk*

n/m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16






27

ground truth

X M z 2 Rm


Characteristic Function m = 60 z = Mp ⇡

KX

k=1

↵kMp✓k

p ⇡KX

k=1

↵kp✓k

Density model=GMM with variance = identity


Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez


[email protected]


Motivation



n x

1

. . .xN


ˆ

A

=) mˆ

z


L=) K ✓






fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)


p =

Pks=1↵sfµs

with:

•weights ↵1


1

, . . . ,µk 2 Rn.


1

N



z =

ˆ


p = argminq2⌃k

kˆz�Aqk22

, (2)



(Rn)


Dictionary {e1



Algorithm





�


argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1


1

, . . . , ↵k.3. Final ”shift”:










1

, . . . ,µk ⇠i.i.d.

N (0, 1).



drawn in0,max

x2X||x||

2


2

(0, 1).




NCompressed EM


3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24


20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size mk*

n/m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16






27

estimated centroids

ground truth

X M z 2 Rm



KX

k=1

↵kMp✓k

p ⇡KX

k=1

↵kp✓k



Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez


[email protected]


Motivation



n x

1

. . .xN


ˆ

A

=) mˆ

z


L=) K ✓






fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)


p =

Pks=1↵sfµs

with:

•weights ↵1


1

, . . . ,µk 2 Rn.


1

N



z =

ˆ


p = argminq2⌃k

kˆz�Aqk22

, (2)



(Rn)


Dictionary {e1



Algorithm





�


argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1


1

, . . . , ↵k.3. Final ”shift”:










1

, . . . ,µk ⇠i.i.d.

N (0, 1).



drawn in0,max

x2X||x||

2


2

(0, 1).




NCompressed EM


3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24


20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size mk*

n/m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16



inspired by

Iterative Hard Thresholding Recovery algorithm

= “decoder”

�




27

estimated centroids

ground truth

X M z 2 Rm



KX

k=1

↵kMp✓k

p ⇡KX

k=1

↵kp✓k



Anthony Bourrier

12, R

´

emi Gribonval

2, Patrick P

´

erez


[email protected]


Motivation



n x

1

. . .xN


ˆ

A

=) mˆ

z


L=) K ✓






fµ / exp

��kx� µk2

2

/(2�2

)

�. (1)


p =

Pks=1↵sfµs

with:

•weights ↵1


1

, . . . ,µk 2 Rn.


1

N



z =

ˆ


p = argminq2⌃k

kˆz�Aqk22

, (2)



(Rn)


Dictionary {e1



Algorithm





�


argmin�2RK

+

||ˆz�U�||22

, (3)

with U = [

ˆµ1


1

, . . . , ↵k.3. Final ”shift”:










1

, . . . ,µk ⇠i.i.d.

N (0, 1).



drawn in0,max

x2X||x||

2


2

(0, 1).




NCompressed EM


3

0.68± 0.28 0.06± 0.01 0.6 0.68± 0.44 0.07± 0.03 0.2410

4

0.24± 0.31 0.02± 0.02 0.6 0.19± 0.21 0.01± 0.02 2.410

5

0.13± 0.15 0.01± 0.02 0.6 0.13± 0.21 0.01± 0.02 24


20, k = 10,m = 1000.

−4 −2 0 2 4 6 8

−4

−2

0

2

4

6

ˆ

A

=)

−0.5 0 0.5 10

10

20

30

40

50

60 n=10, Hell. for 80%

sketch size mk*

n/m

200 400 600 800 1000 1200 1400 1600 1800 2000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16



inspired by

Iterative Hard Thresholding Recovery algorithm

= “decoder”

�


Compressive Hierarchical Splitting (CHS) = extension to general GMM

similar to OMP with Replacement


Application: Speaker Verification Results (DET-curves)

28

MFCC coefficients xi 2 R12

N = 300 000 000

~ 50 Gbytes ~ 1000 hours of speech



28


After silence detection

N = 60 000 000

N = 300 000 000




28



N = 60 000 000

Maximum size manageable by EM

N = 300 000

N = 300 000 000




29



N = 60 000 000


N = 300 000

N = 300 000 000


CHS

for EMfor CHS


CHS


30



N = 60 000 000


N = 300 000

N = 300 000 000


for EM

for CHS



31


m= 500close to EM 7 200 000-fold compression one QR code 40-L

CHS



31


m= 1000same as EM 3 600 000-fold compression two QR codes 40-L


CHS


m= 5 000better than EM exploit whole collection 720 000-fold compression fit 80 on 3”1/2 floppy disk


31


m= 1000same as EM 3 600 000-fold compression two QR codes 40-L


CHS

Computational Efficiency


Computational Aspects

Sketching empirical characteristic function

33

z`

=1

N

NX

i=1

ejw>` xi




33

z`

=1

N

NX

i=1

ejw>` xi

X




33

z`

=1

N

NX

i=1

ejw>` xi

X

W WX

h(WX)

h(·) = ej(·)




33

z`

=1

N

NX

i=1

ejw>` xi

X

W WX

h(WX)

h(·) = ej(·)

z

average




33

z`

=1

N

NX

i=1

ejw>` xi

X

W WX

h(WX)

h(·) = ej(·)

z

average

~ One-layer random neural net DNN ~ hierarchical sketching ?

see also [Bruna & al 2013, Giryes & al 2015]




34

z`

=1

N

NX

i=1

ejw>` xi

X

W WX

h(WX)

h(·) = ej(·)

z

average

Privacy-reserving sketch and forget

~ One-layer random neural net DNN ~ hierarchical sketching ?

see also [Bruna & al 2013, Giryes & al 2015]




35

z`

=1

N

NX

i=1

ejw>` xi

X

W WX

h(WX)

h(·) = ej(·)

z

average

Streaming algorithms One pass; online update




35

z`

=1

N

NX

i=1

ejw>` xi

X

W WX

h(WX)

h(·) = ej(·)

z

average

streaming…

Streaming algorithms One pass; online update




36

z`

=1

N

NX

i=1

ejw>` xi

X

W WX

h(WX)

h(·) = ej(·)

z

average

… … … …

DIS TRI BU TED

Distributed computing Decentralized (HADOOP) / parallel (GPU)

Conclusion


Projections & Learning

38

y

Signal

space

x

Observation space

Signal Processing compressive sensing

M M

Probability space

Sketch space

Machine Learning compressive learning

z

p


Compressive sensing random projections of data items

Compressive learning with sketches

random projections of collections

nonlinear in the feature vectors


Reduce dimension of data items

Reduce size of collection


Summary

Compressive clustering & Compressive GMM Bourrier, G., Perez, Compressive Gaussian Mixture Estimation. ICASSP 2013 Keriven & G.. Compressive Gaussian Mixture Estimation by Orthogonal Matching Pursuit with Replacement. SPARS 2015, Cambridge, United Kingdom Keriven & al, Sketching for Large-Scale Learning of Mixture Models (draft)

Unified framework covering projections & sketches Instance Optimal Decoders Restricted Isometry Property

Bourrier & al, Fundamental performance limits for ideal decoders in high-dimensional linear inverse problems. IEEE Transactions on Information Theory, 2014

39

Challenge: compress before learning ?X

Information preservation ?

Details: poster N. Keriven


Recent / ongoing work / challenges

Sufficient dimension for RIP

Puy, Davies & G., Recipes for stable linear embeddings from Hilbert spaces to ℝ^m, hal-01203614, see also EUSIPCO 2015 and [Dirksen 2014]

RIP for sketches in RKHS applied to compressive GMM upcoming, Keriven, Bourrier, Perez & G.

Compressive statistical learning: intrinsic dimension of PCA and other related learning tasks

work in progress, Blanchard & G.

RIP-based guarantees for general (convex & nonconvex) regularizers

Traonmilin & G, Stable recovery of low-dimensional cones in Hilbert spaces - One RIP to rule them all, arXiv:1510.00504

extends sharp RIP 1/sqrt(2) [Cai & Zhang 2014] beyond sparsity (low-rank; block/structured …)

40

m = O(dB(⌃� ⌃))

Dimension reduction ?

Decoders?

Details: poster G. Puy


•Postdoc / R&D engineer positions @ IRISA ✓ theoretical and algorithmic foundations of large-scale

machine learning & signal processing ✓ funded by ERC project PLEASE

TH###NKS#

Interested ? Joint the team

41

TH###NKS#

Rémi Gribonval Inria Rennes - Bretagne Atlantique · R. GRIBONVAL - CSA 2015 - Berlin Example: Compressive Clustering 9 X Compressive Gaussian Mixture Estimation Anthony Bourrier12,Remi

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