Tensor Methods: A New Paradigm for

Probabilistic Models and Feature Learning

Anima Anandkumar

U.C. Irvine

Learning with Big Data

Data vs. Information

Data vs. Information

Data vs. Information

Missing observations, gross corruptions, outliers.

Data vs. Information

Missing observations, gross corruptions, outliers.

Learning useful information is finding needle in a haystack!

Matrices and Tensors as Data StructuresMulti-modal and multi-relational data.

Matrices: pairwise relations. Tensors: higher order relations.

Multi-modal data figure from Lise Getoor slides.

Spectral Decomposition of Tensors

M2 =

i

iui vi

= + ....

Matrix M2 1u1 v1 2u2 v2

Spectral Decomposition of Tensors

M2 =

i

iui vi

= + ....

Matrix M2 1u1 v1 2u2 v2

M3 =

i

iui vi wi

= + ....

Tensor M3 1u1 v1 w1 2u2 v2 w2

We have developed efficient methods to solve tensor decomposition.

Strengths of Tensor MethodsFast and accurate, orders of magnitude faster than previous methods.

Embarrassingly parallel and suited for cloud systems, e.g.Spark.

Exploit optimized linear algebra libraries.

Exploit parallelism of GPU systems.

Strengths of Tensor MethodsFast and accurate, orders of magnitude faster than previous methods.

Embarrassingly parallel and suited for cloud systems, e.g.Spark.

Exploit optimized linear algebra libraries.

Exploit parallelism of GPU systems.

102 103101

100

101

102

103

104

Number of communities k

Runningtime(secs)

MATLAB Tensor Toolbox(CPU)

CULA Standard Interface(GPU)

CULA Device Interface(GPU)

Eigen Sparse(CPU)

Outline

1 Introduction

2 Learning Probabilistic Models

3 Experiments

4 Feature Learning with Tensor Methods

5 Conclusion

Latent variable models

Incorporate hidden or latent variables.

Information structures: Relationships between latent variables andobserved data.

Latent variable models

Incorporate hidden or latent variables.

Information structures: Relationships between latent variables andobserved data.

Basic Approach: mixtures/clusters

Hidden variable is categorical.

Latent variable models

Incorporate hidden or latent variables.

Information structures: Relationships between latent variables andobserved data.

Basic Approach: mixtures/clusters

Hidden variable is categorical.

Advanced: Probabilistic models

Hidden variables have more general distributions.

Can model mixed membership/hierarchicalgroups.

x1 x2 x3 x4 x5

h1

h2 h3

Challenges in Learning LVMs

Computational Challenges

Maximum likelihood: non-convex optimization. NP-hard.

Practice: Local search approaches such as gradient descent, EM,Variational Bayes have no consistency guarantees.

Can get stuck in bad local optima. Poor convergence rates and hardto parallelize.

Tensor methods yield guaranteed learning for LVMs

Unsupervised Learning of LVMs

GMM HMM

h1 h2 h3

x1 x2 x3

ICA

h1 h2 hk

x1 x2 xd

Multiview and Topic Models

Overall Framework for Unsupervised Learning

= +....

UnlabeledData

Probabilisticadmixturemodels

TensorMethod

Inference

Outline

1 Introduction

2 Learning Probabilistic Models

3 Experiments

4 Feature Learning with Tensor Methods

5 Conclusion

Demo for Learning Gaussian Mixtures

NYTimes Demo

Experimental Results on Yelp and DBLPLowest error business categories & largest weight businesses

Rank Category Business Stars Review Counts1 Latin American Salvadoreno Restaurant 4.0 362 Gluten Free P.F. Changs China Bistro 3.5 553 Hobby Shops Make Meaning 4.5 144 Mass Media KJZZ 91.5FM 4.0 135 Yoga Sutra Midtown 4.5 31

Experimental Results on Yelp and DBLPLowest error business categories & largest weight businesses

Rank Category Business Stars Review Counts1 Latin American Salvadoreno Restaurant 4.0 362 Gluten Free P.F. Changs China Bistro 3.5 553 Hobby Shops Make Meaning 4.5 144 Mass Media KJZZ 91.5FM 4.0 135 Yoga Sutra Midtown 4.5 31

Top-5 bridging nodes (businesses)

Business CategoriesFour Peaks Brewing Restaurants, Bars, American, Nightlife, Food, Pubs, TempePizzeria Bianco Restaurants, Pizza, PhoenixFEZ Restaurants, Bars, American, Nightlife, Mediterranean, Lounges, PhoenixMatts Big Breakfast Restaurants, Phoenix, Breakfast& BrunchCornish Pasty Co Restaurants, Bars, Nightlife, Pubs, Tempe

Experimental Results on Yelp and DBLPLowest error business categories & largest weight businesses

Rank Category Business Stars Review Counts1 Latin American Salvadoreno Restaurant 4.0 362 Gluten Free P.F. Changs China Bistro 3.5 553 Hobby Shops Make Meaning 4.5 144 Mass Media KJZZ 91.5FM 4.0 135 Yoga Sutra Midtown 4.5 31

Top-5 bridging nodes (businesses)

Business CategoriesFour Peaks Brewing Restaurants, Bars, American, Nightlife, Food, Pubs, TempePizzeria Bianco Restaurants, Pizza, PhoenixFEZ Restaurants, Bars, American, Nightlife, Mediterranean, Lounges, PhoenixMatts Big Breakfast Restaurants, Phoenix, Breakfast& BrunchCornish Pasty Co Restaurants, Bars, Nightlife, Pubs, Tempe

Error (E) and Recovery ratio (R)

Dataset k Method Running Time E RDBLP sub(n=1e5) 500 ours 10,157 0.139 89%DBLP sub(n=1e5) 500 variational 558,723 16.38 99%DBLP(n=1e6) 100 ours 5407 0.105 95%

Discovering Gene Profiles of Neuronal Cell Types

Learning mixture of point processes of cells through tensor methods.

Components of mixture are candidates for neuronal cell types.

Discovering Gene Profiles of Neuronal Cell Types

Learning mixture of point processes of cells through tensor methods.

Components of mixture are candidates for neuronal cell types.

Hierarchical Tensors for Healthcare Analytics

= + .... = + .... = +....

= + ....

Hierarchical Tensors for Healthcare Analytics

= + .... = + .... = +....

= + ....

CMS dataset: 1.6 million patients, 15.8 million events.

Mining disease inferences from patient records.

Outline

1 Introduction

2 Learning Probabilistic Models

3 Experiments

4 Feature Learning with Tensor Methods

5 Conclusion

Feature Learning For Efficient Classification

Find good transformations of input for improved classification

Figures used attributed to Fei-Fei Li, Rob Fergus, Antonio Torralba, et al.

Tensor Methods for Training Neural Networks

First order score function m-th order score function

Input pdf is p(x)

Sm(x) :=(1)m(m)p(x)

p(x)

Capture local variations indata.

Algorithm for Training Neural Networks

Estimate score functions using autoencoder.

Decompose tensor E[y Sm(x)] to obtain weights, for m 3.

Recursively estimate score function of autoencoder and repeat.

Tensor Methods for Training Neural Networks

Second order score function m-th order score function

Input pdf is p(x)

Sm(x) :=(1)m(m)p(x)

p(x)

Capture local variations indata.

Algorithm for Training Neural Networks

Estimate score functions using autoencoder.

Decompose tensor E[y Sm(x)] to obtain weights, for m 3.

Recursively estimate score function of autoencoder and repeat.

Tensor Methods for Training Neural Networks

Third order score function m-th order score function

Input pdf is p(x)

Sm(x) :=(1)m(m)p(x)

p(x)

Capture local variations indata.

Algorithm for Training Neural Networks

Estimate score functions using autoencoder.

Decompose tensor E[y Sm(x)] to obtain weights, for m 3.

Recursively estimate score function of autoencoder and repeat.

Demo: Training Neural Networks

Combining Probabilistic Models with Deep Learning

Multi-object Detection in Computer vision

Deep learning is able to extract good features, but not context.

Probabilistic models capture contextual information.

Hierarchical models + pre-trained deep learning features.

State-of-art results on Microsoft COCO.

Outline

1 Introduction

2 Learning Probabilistic Models

3 Experiments

4 Feature Learning with Tensor Methods

5 Conclusion

Conclusion: Tensor Methods for Learning

Tensor Decomposition

Efficient sample and computational complexities

Better performance compared to EM, Variational Bayes etc.

In practice

Scalable and embarrassingly parallel: handle large datasets.

Efficient performance: perplexity or ground truth validation.

My Research Group and Resources

Furong H. Majid J. Hanie S. Niranjan U.N.

Forough A. Tejaswi N. Hao L. Yang S.

ML summer school lectures available athttp://newport.eecs.uci.edu/anandkumar/MLSS.html

IntroductionLearning Probabilistic ModelsExperimentsFeature Learning with Tensor MethodsConclusion