Image Classification using Sparse Coding: Advanced Topics Part 3: Image Classification using Sparse Coding: Advanced Topics Kai Yu Dept. of Media Analytics NEC Laboratories America Andrew Ng Computer Science Dept. Stanford University

Outline of Part 3 Why can sparse coding learn good features? Intuition, topic model view, and geometric view A theoretical framework: local coordinate coding Two practical coding methods Recent advances in sparse coding for image classification 4/19/2017

Outline of Part 3 Why can sparse coding learn good features? Intuition, topic model view, and geometric view A theoretical framework: local coordinate coding Two practical coding methods Recent advances in sparse coding for image classification 4/19/2017

Intuition: why sparse coding helps classification? Figure from http://www.dtreg.com/svm.htm The coding is a nonlinear feature mapping Represent data in a higher dimensional space Sparsity makes prominent patterns more distinctive 4/19/2017

A “topic model” view to sparse coding Basis 1 Basis 2 Both figures adapted from CVPR10 tutorial by F. Bach, J. Mairal, J. Ponce and G. Sapiro Each basis is a “direction” or a “topic”. Sparsity: each datum is a linear combination of only a few bases. Applicable to image denoising, inpainting, and super-resolution. 4/19/2017

A geometric view to sparse coding Data manifold Data Basis Each basis is somewhat like a pseudo data point – “anchor point” Sparsity: each datum is a sparse combination of neighbor anchors. The coding scheme explores the manifold structure of data. 4/19/2017

MNIST Experiment: Classification using SC Try different values 60K training, 10K for test Let k=512 Linear SVM on sparse codes 4/19/2017

MNIST Experiment: Lambda = 0.0005 Each basis is like a part or direction. 4/19/2017

MNIST Experiment: Lambda = 0.005 Again, each basis is like a part or direction. 4/19/2017

MNIST Experiment: Lambda = 0.05 Now, each basis is more like a digit ! 4/19/2017

MNIST Experiment: Lambda = 0.5 Like clustering now! 4/19/2017

Geometric view of sparse coding Error: 4.54% Error: 3.75% Error: 2.64% When SC achieves the best classification accuracy, the learned bases are like digits – each basis has a clear local class association. Implication: exploring data geometry may be useful for classification. 4/19/2017

Distribution of coefficients (MNIST) Neighbor bases tend to get nonzero coefficients Let’s further check what’s happening when best classification performance is achieved. 4/19/2017

Distribution of coefficient (SIFT, Caltech101) Similar observation here! 4/19/2017

Recap: two different views to sparse coding Discover “topic” components Each basis is a “direction” Sparsity: each datum is a linear combination of several bases. Related to topic model View 2 Geometric structure of data manifold Each basis is an “anchor point” Sparsity: each datum is a linear combination of neighbor anchors. Somewhat like a soft VQ (link to BoW) Either can be valid for sparse coding under certain circumstances. View 2 seems to be helpful to sensory data classification. 4/19/2017

Outline of Part 3 Why can sparse coding learn good features? Intuition, topic model view, and geometric view A theoretical framework: local coordinate coding Two practical coding methods Recent advances in sparse coding for image classification 4/19/2017

Key theoretical question Feature learning Classification Why unsupervised feature learning via sparse coding can help classification? 4/19/2017

The image classification setting for analysis Sparse Coding Dense local feature Linear Pooling Linear SVM Implication: Learning an image classifier is a matter of learning nonlinear functions on patches. Function on patches Function on images

Illustration: nonlinear learning via local coding locally linear data points bases 4/19/2017

How to learn a nonlinear function? Step 1: Learning the dictionary from unlabeled data 4/19/2017

How to learn a nonlinear function? Step 2: Use the dictionary to encode data 4/19/2017

How to learn a nonlinear function? Step 3: Estimate parameters Global linear weights to be learned Sparse codes of data Nonlinear local learning via learning a global linear function. 4/19/2017

A good coding scheme should 1. have a small coding error, Local Coordinate Coding (LCC): connect coding to nonlinear function learning Yu et al NIPS-09 If f(x) is (alpha, beta)-Lipschitz smooth The key message: A good coding scheme should 1. have a small coding error, 2. and also be sufficiently local Function approximation error Coding error Locality term 4/19/2017

Outline of Part 3 Why can sparse coding learn good features? Intuition, topic model view, and geometric view A theoretical framework: local coordinate coding Two practical coding methods Recent advances in sparse coding for image classification 4/19/2017

Application of LCC theory Fast Implementation with a large dictionary A simple geometric way to improve BoW Wang et al, CVPR 10 Zhou et al, ECCV 10 4/19/2017

Application of LCC theory Fast Implementation with a large dictionary A simple geometric way to improve BoW 4/19/2017

The larger dictionary, the higher accuracy, but also the higher computation cost Yu et al NIPS-09 Yang et al CVPR 09 The same observation for Caltech-256, PASCAL, ImageNet, … 4/19/2017

Locality-constrained linear coding a fast implementation of LCC Wang et al, CVPR 10 Dictionary Learning: k-means (or hierarchical k-means) Coding for X, Step 1 – ensure locality: find the K nearest bases Step 2 – ensure low coding error: 4/19/2017

Competitive in accuracy, cheap in computation Comparable with sparse coding This is one of the two major algorithms applied by NEC-UIUC team to achieve the No.1 position in ImageNet challenge 2010! Sparse coding Significantly better than sparse coding Wang et al CVPR 10 4/19/2017

Application of the LCC theory Fast Implementation with a large dictionary A simple geometric way to improve BoW 4/19/2017

Interpret “BoW + linear classifier” Piece-wise local constant (zero-order) data points cluster centers

Super-vector coding: a simple geometric way to improve BoW (VQ) Zhou et al, ECCV 10 Piecewise local linear (first-order) Local tangent data points cluster centers

Super-vector coding: a simple geometric way to improve BoW (VQ) If f(x) is beta-Lipschitz smooth, and Local tangent Function approximation error Quantization error 4/19/2017

Super-vector coding: learning nonlinear function via a global linear model Let be the VQ coding of Super-vector codes of data This is one of the two major algorithms applied by NEC-UIUC team to achieve the No.1 position in PASCAL VOC 2009! Global linear weights to be learned 4/19/2017

Summary of Geometric Coding Methods Vector Quantization (BoW) (Fast) Local Coordinate Coding Super-vector Coding All lead to higher-dimensional, sparse, and localized coding All explore geometric structure of data New coding methods are suitable for linear classifiers. Their implementations are quite straightforward.

Things not covered here Improved LCC using Local Tangent, Yu & Zhang, ICML10 Mixture of Sparse Coding, Yang et al ECCV 10 Deep Coding Network, Lin et al NIPS 10 Pooling methods Max-pooling works well in practice, but appears to be ad-hoc. An interesting analysis on max-pooling, Boureau et al. ICML 2010 We are working on a linear pooling method, which has a similar effect as max-pooling. Some preliminary results already in the super-vector coding paper, Zhou et al, ECCV2010. 4/19/2017

Outline of Part 3 Why can sparse coding learn good features? Intuition, topic model view, and geometric view A theoretical framework: local coordinate coding Two practical coding methods Recent advances in sparse coding for image classification 4/19/2017

Fast approximation of sparse coding via neural networks Gregor & LeCun, ICML-10 The method aims at improving sparse coding speed in coding time, not training speed, potentially make sparse coding practical for video. Idea: Given a trained sparse coding model, use its input outputs as training data to train a feed-forward model They showed a speedup of X20 faster. But not evaluated on real video data. 4/19/2017

Group sparse coding Bengio et al, NIPS 09 Sparse coding is on patches, the image representation is unlikely sparse. Idea: enforce joint sparsity via L1/L2 norm on sparse codes of a group of patches. The resultant image representation becomes sparse, which can save the memory cost, but the classification accuracy decreases. 4/19/2017

Learning hierarchical dictionary Jenatton, Mairal, Obozinski, and Bach, 2010 A node can be active only if its ancestors are active. 4/19/2017

Reference Image Classification using Super-Vector Coding of Local Image Descriptors, Xi Zhou, Kai Yu, Tong Zhang, and Thomas Huang. In ECCV 2010. Efficient Highly Over-Complete Sparse Coding using a Mixture Model, Jianchao Yang, Kai Yu, and Thomas Huang. In ECCV 2010. Learning Fast Approximations of Sparse Coding, Karol Gregor and Yann LeCun. In ICML 2010. Improved Local Coordinate Coding using Local Tangents, Kai Yu and Tong Zhang. In ICML 2010. Sparse Coding and Dictionary Learning for Image Analysis, Francis Bach,  Julien Mairal, Jean Ponce, and Guillermo Sapiro. CVPR 2010 Tutorial Supervised translation-invariant sparse coding, Jianchao Yang, Kai Yu, and Thomas Huang, In CVPR 2010. Learning locality-constrained linear coding for image classification, Jingjun Wang, Jianchao Yang, Kai Yu, Fengjun Lv, Thomas Huang, and Yihong Gong. In CVPR 2010. Group Sparse Coding, Samy Bengio, Fernando Pereira, Yoram Singer, and Dennis  Strelow, In NIPS*2009. Nonlinear learning using local coordinate coding, Kai Yu, Tong Zhang, and Yihong Gong. In NIPS*2009. Linear spatial pyramid matching using sparse coding for image classification, Jianchao Yang, Kai Yu, Yihong Gong, and Thomas Huang. In CVPR 2009. Efficient sparse coding algorithms. Honglak Lee, Alexis Battle, Raina Rajat and Andrew Y.Ng. In NIPS*2007. 4/19/2017