Dictionary learning of sound speed profiles Michael Bianco a) and Peter Gerstoft Scripps Institution of Oceanography, University of California San Diego, La Jolla, California 92093–0238, USA (Received 21 August 2016; revised 6 January 2017; accepted 17 February 2017; published online 13 March 2017) To provide constraints on the inversion of ocean sound speed profiles (SSPs), SSPs are often mod- eled using empirical orthogonal functions (EOFs). However, this regularization, which uses the lead- ing order EOFs with a minimum-energy constraint on the coefficients, often yields low resolution SSP estimates. In this paper, it is shown that dictionary learning, a form of unsupervised machine learning, can improve SSP resolution by generating a dictionary of shape functions for sparse proc- essing (e.g., compressive sensing) that optimally compress SSPs; both minimizing the reconstruction error and the number of coefficients. These learned dictionaries (LDs) are not constrained to be orthogonal and thus, fit the given signals such that each signal example is approximated using few LD entries. Here, LDs describing SSP observations from the HF-97 experiment and the South China Sea are generated using the K-SVD algorithm. These LDs better explain SSP variability and require fewer coefficients than EOFs, describing much of the variability with one coefficient. Thus, LDs improve the resolution of SSP estimates with negligible computational burden. C 2017 Acoustical Society of America . [http://dx.doi.org/10.1121/1.4977926] V [ZHM] Pages: 1749–1758 I. INTRODUCTION shape functions such as wavelets or curvelets, however exten- sive analysis is required to find an optimal set of prescribed Inversion for ocean sound speed profiles (SSPs) using shape functions. Dictionary learning proposes a more direct acoustic data is a non-linear and highly underdetermined approach: given enough signal examples for a given signal problem. 1 To ensure physically realistic solutions while class, learn a dictionary of shape functions that approximate moderating the size of the parameter search, SSP inversion signals within the class using few coefficients. These learned has often been regularized by modeling SSP as the sum of dictionaries (LDs) have improved compression and denoising leading order empirical orthogonal functions (EOFs). 2–7 results for image and video data over ad hoc dictionaries. 9,11 However, regularization using EOFs often yields low resolu- Dictionary learning has been applied to denoising problems tion estimates of ocean SSPs, which can be highly variable in seismics 25 and ocean acoustics, 26,27 as well as to structural with fine scale fluctuations. In this paper, it is shown that the acoustic health monitoring. 28 resolution of SSP estimates are improved using dictionary The K-SVD algorithm, 12 a popular dictionary learning learning, 8–13 a form of unsupervised machine learning, to method, finds a dictionary of vectors that optimally partition generate a dictionary of regularizing shape functions from the data from the training set such that the few dictionary vec- SSP data for parsimonious representation of SSPs. tors describe each data example. Relative to EOFs which are Many signals, including natural images, 14,15 audio, 16 derived using principal component analysis (PCA), 29,30 these and seismic profiles 17 are well approximated using sparse LDs are not constrained to be orthogonal. Thus, LD’s provide (few) coefficients, provided a dictionary of shape functions potentially better signal compression because the vectors are exist under which their representation is sparse. Given a on average, nearer to the signal examples (see Fig. 1). 13 K -dimensional signal, a dictionary is defined as a set of N , In this paper, LDs describing one dimensional (1D) ‘ 2 -normalized vectors which describe the signal using few ocean SSP data from the HF-97 experiment, 31,32 and from coefficients. The sparse processor is then an ‘ 2 -norm cost the South China Sea (SCS) 33 are generated using the K-SVD function with an ‘ 0 -norm penalty on the number of non- algorithm and the reconstruction performance is evaluated zero coefficients. Signal sparsity is exploited for a number against EOF methods. In Sec. II, EOFs, sparse reconstruction of purposes including signal compression and denoising. 9 methods, and compression are introduced. In Sec. III, the Applications of compressive sensing, 18 one approximation K-SVD dictionary learning algorithm is explained. In Sec. to the ‘ 0 -norm sparse processor, have in ocean acoustics shown improvements in beamforming, 19–22 geoacoustic IV, SSP reconstruction results are given for LDs and EOFs. It inversion, 23 and estimation of ocean SSPs. 24 is shown that each shape function within the resulting LDs explain more SSP variability than the leading order EOFs Dictionaries that approximate a given class of signals trained on the same data. Further, it is demonstrated that using few coefficients can be designed using dictionary learn- SSPs can be reconstructed up to acceptable error using as few ing. 9 Dictionaries can be generated ad hoc from common as one non-zero coefficient. This compression can improve the resolution of ocean SSP estimates with negligible compu- a) Electronic mail: mbianco@ucsd.edu tational burden. C 2017 Acoustical Society of America J. Acoust. Soc. Am. 141 (3), March 2017 0001-4966/2017/141(3)/1749/10/$30.00 1749 V
value of the M original observations is subtracted to obtain Y . The variance of the SSP anomaly at each depth sample k , r 2 k , is defined as M k ¼ 1 � 2 ; X r 2 � y k (1) m M m ¼ 1 where ½ y k 1 ; … ; y k M � are the SSP anomaly values at depth sam- ple k for M time samples. The singular value decomposition (SVD) 34 finds the EOFs as the eigenvectors of YY T by YY T ¼ P K 2 P T ; (2) where P ¼ ½ p 1 ; … ; p L � 2 R K � L are EOFs (eigenvectors) and K 2 ¼ diag ð½ k 2 L �Þ 2 R L � L are the total variances of the 1 ; … ; k 2 data along the principal directions defined by the EOFs p l with K k ¼ 1 X ð Þ : r 2 M tr K 2 (3) k ¼ 1 The EOFs p l with k 2 1 � � � � � k 2 L are spatial features of the SSPs which explain the greatest variance of Y . If the number of training vectors M � K , L ¼ K and [ p 1 ,…, p L ] form a basis in R K . B. SSP reconstruction using EOFs Since the leading-order EOFs often explain much of the variance in Y , the representation of anomalies y m can be compressed by retaining only the leading order EOFs P < L , FIG. 1. (Color online) (a) EOF vectors [ u 1 , u 2 ] and (b) overcomplete LD vectors [ q 1 ,…, q N ] for arbitrary 2D Gaussian distribution relative to arbitrary y m ¼ Q P ^ ^ x P ; m ; (4) 2D data observation y m . where Q P 2 R K � P is here the dictionary containing the P Notation : In the following, vectors are represented by x P ; m 2 R P is the coefficient vector. leading-order EOFs and ^ bold lower-case letters and matrices by bold uppercase Since the entries in Q P are orthonormal, the coefficients are letters. The ‘ p -norm of the vector x 2 R N is defined solved by k x k p ¼ ð P N n ¼ 1 j x n j p Þ 1 = p . as Using similar notation, the n ¼ 1 j x n j 0 ¼ P N x P ; m ¼ Q T ‘ 0 -norm is defined as k x k 0 ¼ P N ^ P y m : (5) n ¼ 1 1 j x n j > 0 . The ‘ p -norm of the matrix A 2 R K � M is defined as k A k p For ocean SSPs, usually no more than P ¼ 5 EOF coeffi- ¼ ð P M P K k j p Þ 1 = p . The Frobenius norm ( ‘ 2 -norm) of k ¼ 1 j a m cients have been used to reconstruct ocean SSPs. 4,7 m ¼ 1 the matrix A is written as k A k F . The hat symbol ^ appearing above vectors and matrices indicates approximations to the C. Sparse reconstruction true signals or coefficients. A signal y m , whose model is sparse in the dictionary Q N ¼ ½ q 1 ; … ; q N � 2 R K � N ( N -entry sparsifying dictionary II. EOFS AND COMPRESSION for Y ), is reconstructed to acceptable error using T � K vec- tors q n . 9 The problem of estimating few coefficients in x m A. EOFs and PCA for reconstruction of y m can be phrased using the canonical Empirical orthogonal function (EOF) analysis seeks to sparse processor reduce the dimension of continuously sampled space-time fields by finding spatial patterns which explain much of the ^ x m ¼ arg min x m 2 R N k y m � Qx m k 2 subject to k x m k 0 � T : (6) variance of the process. These spatial patterns or EOFs cor- respond to the principal components, from principal compo- nent analysis (PCA), of the temporally varying field. 29 Here, The ‘ 0 -norm penalizes the number of non-zero coefficients in the solution to a typical ‘ 2 -norm cost function. The ‘ 0 - the field is a collection of zero-mean ocean SSP anomaly vectors Y ¼ ½ y 1 ; … ; y M � 2 R K � M , which are sampled over K norm constraint is non-convex and imposes combinatorial discrete points in depth and M instants in time. The mean search for the exact solution to Eq. (6). Since exhaustive 1750 J. Acoust. Soc. Am. 141 (3), March 2017 Michael Bianco and Peter Gerstoft
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