Learning Measurement Matrices for Redundant Dictionaries Richard Baraniuk Rice University Chinmay Hegde MIT Aswin Sankaranarayanan CMU

Sparse Recovery Sparsity rocks, etc. Previous talk focused mainly on signal inference (ex: classification, NN search) This talk focuses on signal recovery

Compressive Sensing Sensing via randomized dimensionality reduction random measurements sparse signal nonzero entries Recovery:solve an ill-posed inverse problem exploit the geometrical structure of sparse/compressible signals

Gaussian measurements incoherent with any fixed orthonormal basis (with high probability) Ex: frequency domain: General Sparsifying Bases

Sparse Modeling: Approach 1 Step 1: Choose a signal model with structure –e.g. bandlimited, smooth with r vanishing moments, etc. Step 2: Analytically design a sparsifying basis/frame that exploits this structure –e.g. DCT, wavelets, Gabor, etc. DCT Wavelets Gabor ?

Sparse Modeling: Approach 2 Learn the sparsifying basis/frame from training data Problem formulation: given a large number of training signals, design a dictionary D that simultaneously sparsifies the training data Called sparse coding / dictionary learning

Dictionaries Dictionary: an NxQ matrix whose columns are used as basis functions for the data Convention: assume columns are unit-norm More columns than rows, so dictionary is redundant / overcomplete

Dictionary Learning Rich vein of theoretical and algorithmic work Olshausen and Field [‘97], Lewicki and Sejnowski [’00], Elad [‘06], Sapiro [‘08] Typical formulation: Given training data Solve: Several efficient algorithms, ex: K-SVD

Dictionary Learning Successfully applied to denoising, deblurring, inpainting, demosaicking, super-resolution, … –State-of-the-art results in many of these problems Aharon and Elad ‘06

Dictionary Coherence Suppose that the learned dictionary is normalized to have unit -norm columns: The mutual coherence of D is defined as Geometrically, represents the cosine of the minimum angle between the columns of D, smaller is better Crucial parameter in analysis as well as practice (line of work starting with Tropp [04])

Dictionaries and CS Can extend CS to work with non-orthonormal, redundant dictionaries Coherence of determines recovery success Rauhut et al. [08], Candes et al. [10] Fortunately, random guarantees low coherence Holographic basis

Geometric Intuition Columns of D: points on the unit sphere Coherence: minimum angle between the vectors J-L Lemma: Random projections approximately preserve angles between vectors

Q: Can we do better than random projections for dictionary-based CS? Q restated: For a given dictionary D, find the best CS measurement matrix

Optimization Approach Assume that a good dictionary D has been provided. Goal: Learn the best for this particular D As before, want the “shortest” matrix such that the coherence of is at most some parameter To avoid degeneracies caused by a simple scaling, also want that does not shrink columns much:

A NuMax-like Framework Convert quadratic constraints in into linear constraints in (via the “lifting trick”) Use a nuclear-norm relaxation of the rank Simplified problem:

Alternating Direction Method of Multipliers (ADMM) - solve for P using spectral thresholding - solve for L using least-squares - solve for q using “squishing” Convergence rate depends on the size of the dictionary (since #constraints = ) Algorithm: “NuMax-Dict” [HSYB12]

NuMax vs. NuMax-Dict Same intuition, trick, algorithm, etc; Key enabler is that coherence is intrinsically a quadratic function of the data Key difference: the (linearized) constraints are no longer symmetric –We have constraints of the form –This might result in intermediate P estimates having complex eigenvalues, so the notion of spectral thresholding needs to be slightly modified

Experimental Results

Expt 1: Synthetic Dictionary Generic dictionary: random w/ unit norm. columns Dictionary size: 64x128 We construct different measurement matrices: Random NuMax-Dict Algorithm by Elad [06] Algorithm by Duarte-Carvajalino & Sapiro [08] We generate K=3 sparse signals with Gaussian amplitudes, add 30dB measurement noise Recovery using OMP Measure recovery SNR, plot as a function of M

Exp 1: Synthetic Dictionary

Expt 2: Practical Dictionaries 2x overcomplete DCT dictionary, same parameters 2x overcomplete dictionary learned on 8x8 patches of a real-world image (Barbara) using K-SVD Recovery using OMP

Analysis Exact problem seems to be hard to analyze But, as in NuMax, can provide analytical bounds in the special case where the measurement matrix is further constrained to be orthonormal

Orthogonal Sensing of Dictionary-Sparse Signals Given a dictionary D, find the orthonormal measurement matrix that provides the best possible coherence From a geometric perspective, ortho-projections cannot improve coherence, so necessarily

Semidefinite Relaxation The usual trick: Lifting and trace-norm relaxation

Theoretical Result Theorem: For any given redundant dictionary D, denote its mutual coherence by. Denote the optimum of the (nonconvex) problem as Then, there exists a method to produce a rank-2M ortho matrix such that the coherence of is at most i.e., We can obtain close to optimal performance, but pay a price of a factor 2 in the number of measurements

Conclusions NuMax-Dict performance comparable to the best existing algorithms Principled convex optimization framework Efficient ADMM-type algorithm that exploits the rank-1 structure of the problem Upshot: possible to incorporate other structure into the measurement matrix, such as positivity, sparsity, etc.

Open Question Above framework assumes a two-step approach: first construct a redundant dictionary (analytically or from data) and then construct a measurement matrix Given a large number of training data, how to efficiently solve jointly for both the dictionary and the sensing matrix? (Approach introduced in DC-Sapiro [08])