### abstract ###
This article treats the problem of learning a dictionary providing sparse representations for a given signal class, via  SYMBOL -minimisation
The problem can also be seen as factorising a  SYMBOL  matrix  SYMBOL  of training signals into a  SYMBOL  dictionary matrix  SYMBOL  and a  SYMBOL  coefficient matrix   SYMBOL , which is sparse
The exact question studied here is when a dictionary coefficient pair  SYMBOL  can be recovered as local minimum of a (nonconvex)  SYMBOL -criterion with input  SYMBOL
First, for general dictionaries and coefficient matrices, algebraic conditions ensuring local identifiability  are derived, which are then specialised to the case when the dictionary is a basis
Finally, assuming a random Bernoulli-Gaussian sparse model on the coefficient matrix, it is shown that sufficiently incoherent bases are locally identifiable with high probability
The perhaps surprising result is that the typically sufficient number of training samples  SYMBOL  grows up to a logarithmic factor only linearly with the signal dimension, ie SYMBOL , in contrast to previous approaches requiring combinatorially many samples
### introduction ###
Many signal processing tasks, such as denoising and compression, can be efficiently performed if one knows a sparse representation of the signals of interest
Moreover, a huge body of recent results on sparse representations has highlighted their impact on inverse linear problems such as (blind) source separation and localisation as well as compressed sampling, for a starting point see eg CITATION \\ In any of these publications, one will - more likely than not - find a statement starting with 'given a dictionary  SYMBOL  and a signal  SYMBOL  having an  SYMBOL -sparse approximation/representation  SYMBOL  \ldots', which points exactly to the remaining problem: all applications of sparse representations rely on a signal dictionary  SYMBOL  from which sparse linear expansions can be built that efficiently approximate the signals from a class of interest; success heavily depends on the good fit between the data class and the dictionary \\ For many signal classes, good dictionaries -- such as time-frequency or time-scale dictionaries -- are known, but new data classes may require the construction of new dictionaries to fit new types of data features
The analytic construction of dictionaries such as wavelets and curvelets stems from deep  mathematical tools from Harmonic Analysis
It may, however, be difficult and time consuming to develop complex mathematical theory each time a new class of data, which requires a different type of dictionary,  is met
An alternative approach is dictionary learning, which aims at infering the dictionary  SYMBOL  from a set of training data  SYMBOL
Dictionary learning, also known as  sparse coding , has the potential of 'industrialising' sparse representation techniques for new data classes \\ This article treats the theoretical dictionary learning problem, expressed as a factorisation problem which consists of identifying a  SYMBOL  matrix  SYMBOL  from a set of  SYMBOL  observed training vectors  SYMBOL ,  knowing that  SYMBOL ,  SYMBOL  for some unknown collection of coefficient vectors  SYMBOL  with certain statistical properties \\ Considering the extensive literature available for the sparse decomposition problem after the early work in ~ CITATION , surprisingly little work has been dedicated to theoretical dictionary learning so far
There exist several dictionary learning algorithms (see eg CITATION ), but only recently people have started to consider also the theoretical aspects of the problem
The origins of research into what is now called dictionary learning can be found in the field of Independent Component Analysis (ICA)~ CITATION
There, many identifiability results are available, which, however, rely on  asymptotic  statistical properties under  statistical independence  and  non-Gaussianity  assumptions \\ In contrast, Georgiev, Theis and Cichocki,  CITATION , as well as Aharon, Elad and Bruckstein,  CITATION , described more geometric identifiability conditions on the sparse coefficients of training data in an ideal (overcomplete) dictionary
Yet, for these conditions to hold, the size  SYMBOL  of the training set seems to be required to grow exponentially fast with the number of atoms  SYMBOL , and the provably good identification algorithms are combinatorial
Moreover, the algorithms and the identifiability analysis are not robust to 'outliers', i e , training samples  SYMBOL  where  SYMBOL  fails to be sufficiently sparse
For applications, on the other hand, we are concerned with relatively large-dimensional data (e g SYMBOL , or even  SYMBOL ) but limited availability of training data ( SYMBOL  is not much larger than say  SYMBOL ) as well as limited computational resources \\ In this article, we study the possibility of designing provably good, non-combinatorial dictionary learning algorithms that are robust to outliers and to the limited availability of training samples
Inspired by recent proofs of good properties of  SYMBOL -minimisation for sparse signal decomposition with a given dictionary, we investigate the properties of  SYMBOL -based dictionary learning,  CITATION
Our ultimate goal, described in details in Section~, is to characterise properties that a set of training samples  SYMBOL  should satisfy to guarantee that an ideal dictionary is the only local minimum of the  SYMBOL -criterion, opening up the possibility of replacing combinatorial learning algorithms with efficient numerical descent techniques
As a first step, we investigate conditions under which an ideal dictionary is a local minimum of the  SYMBOL -criterion \\ {Main results }  First,  we describe the proposed setting in Section~ and characterise the local minima of the  SYMBOL -cost function in Section~
We discuss the geometrical interpretation of this characterisation in Section~
Then, using concentration of measure, we prove in Section~ the perhaps surprising result that when    SYMBOL  if the samples  SYMBOL , are a typical draw from a Bernoulli-Gaussian random distribution (which can generate a large proportion of  outliers ), then any sufficiently incoherent basis matrix  SYMBOL ,  SYMBOL , is a local minimum of the cost function and is therefore 'locally identifiable'
The constant  SYMBOL  depends on a parameter of the Bernoulli-Gaussian distribution which drives the sparsity of the training set \\ This number of training samples is surprisingly small considering that  SYMBOL  training samples provide  SYMBOL  real parameters, while the basis matrix  SYMBOL  is essentially parameterised by  SYMBOL  independent real parameters \\ In the considered matrix identification setting, it should be noted that  SYMBOL  is  not  a convex cost function
It admits  several local minima  hence local identifiability only implies that, upon good initial conditions, numerical optimisation schemes performing the  SYMBOL -optimisation will recover the desired matrix  SYMBOL
However, empirical experiments in low dimension ( SYMBOL ), shown in Section~, indicate that for typical draws of Bernoulli-Gaussian training samples  SYMBOL , the matrix  SYMBOL  is in fact the  only  local minimum of the criterion (up to natural indeterminacies of the problem such as column permutation)
If this empirical observation could be turned into a theorem for general dimension  SYMBOL  under the Bernoulli-Gaussian sparse model, this would imply that typically: a)  SYMBOL -minimisation is a good  identification principle ; b) any decent  SYMBOL -descent algorithm is a good  identification algorithm
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tex                                                                                          0000644 0000000 0000000 00000005040 11311243514 011564  0                                                                                                    ustar   root                            root                                                                                                                                                                                                                   % \newcommand\dico{\mathbf{\Phi}} \newcommand\atom{\varphi} \newcommand\inn[2]{\langle\atom_{#1}, \atom_{#2}\rangle} \newcommand\ip[2]{#1, #2\rangle} \newcommand\natoms{K} \newcommand\sparsity{S} \newcommand\ddim{d} \newcommand\good{\Lambda} \newcommand\bad{{\overline{\good}}} \newcommand\sensing{\mathbf{\Psi}} \newcommand\satom{\psi} \newcommand\ident{\mathbf{I}} \newcommand\Id{\mathbf{I}} \newcommand\Proj{\mathbf{P}} \newcommand\Q{\mathbf{Q}} \newcommand\Gram{\mathbf{G}} \newcommand\DX{\Sigma} %diagonal matrix with norms of coefficients \newcommand\coeff{\sigma} \newcommand\Dynamic{\operatorname{R}} \newcommand\PSNR{\operatorname{PSNR}} \newcommand\samesim{\beta} \newcommand\gwJ{{\goodJ}} \newcommand\noise{{\eta}} \newcommand\ie{{i e }} \newcommand\iid{{ iid  }} \newcommand\Y{Y} \newcommand\X{X} \newcommand\rownormX[1]{\mathcal{X}^{#1}} \newcommand\eps{\varepsilon} \newcommand\manifold{\mathcal{D}} \newcommand\tangentspace[2]{T_{#2}#1} \newcommand\Sph{\mathcal{S}} \newcommand\pnorm{{q}} \newcommand\randsign[0]{\xi} \def\radius{R} {\mathbf{\Psi}} {\psi} {\mathbf{M}_0} {\mathbf{D}_0} {\mathbf{V}} {\mathbf{Z}} {\mathbf{\Delta}} {\mathbf{A}} \newcommand\Ogroup[1]{\mathcal{O}(#1)} \newcommand\Uclass[1]{\mathcal{B}(#1)} \newcommand\nsig[0]{N} \newcommand\Xbar[0]{\bar{\X}} \newcommand\Cost[1]{\mathcal{C}_{#1}} \newcommand\Null{\mathcal{N}} \newcommand\Nvec{\mathbf{N}} \newcommand\epscover{\mathcal{X}}  \newcommand\U{U} \newcommand\Err{E} \newcommand\diag{\operatorname{diag}} \newcommand\signop{\operatorname{sign}} \newcommand\sign{\operatorname{sign}} \newcommand\trace{\operatorname{trace}} \newcommand\B{\mathbf{B}} \newcommand{\N}{{\mathbb{N}}} \newcommand{\R}{{\mathbb{R}}} \newcommand{\Z}{{\mathbb{Z}}} \newcommand{\C}{{\mathbb{C}}} \renewcommand{\P}{{\mathbb{P}}} \newcommand{\E}{{\mathbb{E}}} \newcommand\opnorm[1]{|\ |\ | #1|\ |\ |}  \newcommand{\rec}{{\operatorname{recovery}}} \newcommand{\nrec}{{\operatorname{not~recovery}}}  \newtheorem{Theorem}{Theorem}[section] \newtheorem{lemma}[Theorem]{Lemma} \newtheorem{corollary}[Theorem]{Corollary} \newtheorem{example}[Theorem]{Example} \newtheorem{proposition}[Theorem]{Proposition} \newtheorem{definition}{Definition}[section] \newtheorem{conj}[Theorem]{Conjecture}  \newtheorem{remark}{Remark}[section]  \newenvironment{Proof}{ {\bf\underline{Proof:} }} {\hspace*{\fill} SYMBOL \vskip1em}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 phasetrans
eps                                                                                      0000644 0000000 0000000 00000031605 11325617152 012445  0                                                                                                    ustar   root                            root                                                                                                                                                                                                                   %
PS-Adobe-3 0 EPSF-3 0  /MathWorks 160 dict begin /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef /c  /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def /dpi2point 0 def /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef /SO { [] 0 setdash } bdef /DO { [ 5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [ 5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup  0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop	translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc  tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix  smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile  3 index 0 eq {/ASCIIHexDecode filter} {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } ifelse exch readstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform  rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def  MathWorks begin  0 cap  end  MathWorks begin bpage  bplot  /dpi2point 12 def portraitMode 1164 5808 csm  0     0  4800  1536 rc 85 dict begin %Colortable dictionary /c0 { 0 000000 0 000000 0 000000 sr} bdef /c1 { 1 000000 1 000000 1 000000 sr} bdef /c2 { 0 900000 0 000000 0 000000 sr} bdef /c3 { 0 000000 0 820000 0 000000 sr} bdef /c4 { 0 000000 0 000000 0 800000 sr} bdef /c5 { 0 910000 0 820000 0 320000 sr} bdef /c6 { 1 000000 0 260000 0 820000 sr} bdef /c7 { 0 000000 0 820000 0 820000 sr} bdef c0 1 j 1 sg 0    0 4801 1537 rf 6 w 0 897 898 0 0 -897 750 1189 4 MP PP -898 0 0 897 898 0 0 -897 750 1189 5 MP stroke gs 750 292 899 898 rc /mwscm { [/Indexed /DeviceRGB 63 < 000000 040404 080808 0c0c0c 101010 141414 181818 1c1c1c 202020 242424  282828 2c2c2c 303030 343434 383838 3c3c3c 404040 444444 484848 4c4c4c  505050 555555 595959 5d5d5d 616161 656565 696969 6d6d6d 717171 757575  797979 7d7d7d 818181 858585 898989 8d8d8d 919191 959595 999999 9d9d9d  a1a1a1 a5a5a5 aaaaaa aeaeae b2b2b2 b6b6b6 bababa bebebe c2c2c2 c6c6c6  cacaca cecece d2d2d2 d6d6d6 dadada dedede e2e2e2 e6e6e6 eaeaea eeeeee  f2f2f2 f6f6f6 fafafa ffffff  > ] setcolorspace } bdef mwscm gs np 751 292 mt 0 898 rl 897 0 rl 0 -898 rl cp c np [897 0 0 898 751 292] cc << % Image dictionary /ImageType 1 /Width 9 /Height 9 /BitsPerComponent 8 /Decode [0 255] /ImageMatrix [9 000000 0 0 9 000000 0 0] /DataSource currentfile /ASCII85Decode filter /RunLengthDecode filter >> image ,7+oG SYMBOL n*,10,GWA
*KI)`8]c+8u6D&/R&A0c0cg":Q\/0,=Bb
spA&)uBU:  SYMBOL Oe=;55mbk) SYMBOL kNar3^3`I
XoYB SYMBOL XJ, ~> gr gr  0 sg 1910  197 mt  (Spurious local minimum) s /Symbol /ISOLatin1Encoding 120 FMSR  2511 1472 mt  (m) s /Helvetica /ISOLatin1Encoding 120 FMSR  1816  775 mt  -90 rotate (p) s 90 rotate 4 w DO SO 6 w 2099 1189 mt 2996 1189 L 2099  292 mt 2996  292 L 2099 1189 mt 2099  292 L 2996 1189 mt 2996  292 L 2099 1189 mt 2996 1189 L 2099 1189 mt 2099  292 L 2248 1189 mt 2248 1180 L 2248  292 mt 2248  301 L 2165 1335 mt  (0 2) s 2447 1189 mt 2447 1180 L 2447  292 mt 2447  301 L 2364 1335 mt  (0 4) s 2647 1189 mt 2647 1180 L 2647  292 mt 2647  301 L 2564 1335 mt  (0 6) s 2846 1189 mt 2846 1180 L 2846  292 mt 2846  301 L 2763 1335 mt  (0 8) s 2099 1040 mt 2107 1040 L 2996 1040 mt 2987 1040 L 1898 1084 mt  (0 2) s 2099  840 mt 2107  840 L 2996  840 mt 2987  840 L 1898  884 mt  (0 4) s 2099  641 mt 2107  641 L 2996  641 mt 2987  641 L 1898  685 mt  (0 6) s 2099  442 mt 2107  442 L 2996  442 mt 2987  442 L 1898  486 mt  (0 8) s 2099 1189 mt 2996 1189 L 2099  292 mt 2996  292 L 2099 1189 mt 2099  292 L 2996 1189 mt 2996  292 L 1 sg 0 897 897 0 0 -897 3447 1189 4 MP PP -897 0 0 897 897 0 0 -897 3447 1189 5 MP stroke gs 3447 292 898 898 rc /mwscm { [/Indexed /DeviceRGB 63 < 000000 040404 080808 0c0c0c 101010 141414 181818 1c1c1c 202020 242424  282828 2c2c2c 303030 343434 383838 3c3c3c 404040 444444 484848 4c4c4c  505050 555555 595959 5d5d5d 616161 656565 696969 6d6d6d 717171 757575  797979 7d7d7d 818181 858585 898989 8d8d8d 919191 959595 999999 9d9d9d  a1a1a1 a5a5a5 aaaaaa aeaeae b2b2b2 b6b6b6 bababa bebebe c2c2c2 c6c6c6  cacaca cecece d2d2d2 d6d6d6 dadada dedede e2e2e2 e6e6e6 eaeaea eeeeee  f2f2f2 f6f6f6 fafafa ffffff  > ] setcolorspace } bdef mwscm gs np 3447 292 mt 0 898 rl 897 0 rl 0 -898 rl cp c np [897 0 0 898 3447 292] cc << % Image dictionary /ImageType 1 /Width 9 /Height 9 /BitsPerComponent 8 /Decode [0 255] /ImageMatrix [9 000000 0 0 9 000000 0 0] /DataSource currentfile /ASCII85Decode filter /RunLengthDecode filter >> image "Z\_14Y/Ji57T%q4=q[55
M7L)_3Td55mbk'bD'#+[%qP56_Au SYMBOL [(+
B8u <F<n ~> gr gr  0 sg 3266  197 mt  (Wrong global mimimum) s /Symbol /ISOLatin1Encoding 120 FMSR  3859 1472 mt  (m) s /Helvetica /ISOLatin1Encoding 120 FMSR  3164  775 mt  -90 rotate (p) s 90 rotate 4 w DO SO 6 w 3447 1189 mt 4344 1189 L 3447  292 mt 4344  292 L 3447 1189 mt 3447  292 L 4344 1189 mt 4344  292 L 3447 1189 mt 4344 1189 L 3447 1189 mt 3447  292 L 3596 1189 mt 3596 1180 L 3596  292 mt 3596  301 L 3513 1335 mt  (0 2) s 3795 1189 mt 3795 1180 L 3795  292 mt 3795  301 L 3712 1335 mt  (0 4) s 3995 1189 mt 3995 1180 L 3995  292 mt 3995  301 L 3912 1335 mt  (0 6) s 4194 1189 mt 4194 1180 L 4194  292 mt 4194  301 L 4111 1335 mt  (0 8) s 3447 1040 mt 3455 1040 L 4344 1040 mt 4335 1040 L 3246 1084 mt  (0 2) s 3447  840 mt 3455  840 L 4344  840 mt 4335  840 L 3246  884 mt  (0 4) s 3447  641 mt 3455  641 L 4344  641 mt 4335  641 L 3246  685 mt  (0 6) s 3447  442 mt 3455  442 L 4344  442 mt 4335  442 L 3246  486 mt  (0 8) s 3447 1189 mt 4344 1189 L 3447  292 mt 4344  292 L 3447 1189 mt 3447  292 L 4344 1189 mt 4344  292 L  end %%Color Dict  eplot  epage end  showpage                                                                                                                             phasetrans
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