Analysis of weighted $\ell_1$-minimization for model based compressed sensing

0.25 MB·English

by  Sidhant Misra

1
ℓ
Analysis of weighted -minimization for
1
model-based compressed sensing
Sidhant Misra and Pablo A. Parrilo
Abstract
3 Model-basedcompressedsensingreferstocompressedsensingwithextrastructureabouttheunderlying
1 sparse signal known a priori. Recent work has demonstrated that both for deterministic and probabilistic
0 models imposed on the signal, this extra information can be successfully exploited to enhance recovery
2 performance. In particular, weighted ℓ1-minimization with suitable choice of weights has been shown to
improveperformanceatleastfor asimpleclassof probabilisticmodels. Inthispaper, weconsider amore
n
general and natural class of probabilistic models where the underlying probabilities associated with the
a
indices of the sparse signal have a continuously varying nature. We prove that when the measurements
J
are obtained using a matrix with i.i.d Gaussian entries, weighted ℓ1-minimization with weights that have
7
a similar continuously varying nature successfully recovers the sparse signal from its measurements with
overwhelming probability. It is known that standard ℓ1-minimization with uniform weights can recover
]
T sparse signals up to a known sparsity level, or expected sparsity level in case of a probabilistic signal
model.Withsuitablechoiceofweightswhicharechosenbasedonoursignalmodel,weshowthatweighted
.I ℓ1-minimization can recover signals beyond the sparsity level achievable by standard ℓ1-minimization.
s
c
[
I. INTRODUCTION

1
CompressedSensinghasemergedasamodernalternativetotraditionalundersamplingforcompressible
v
signals. Previously, the most common way to view recovery of signals from samples was based on the
7
2 Nyquist criterion. According to the Nyquist criterion, a band-limited signal has to be sampled at a rate
3 at least twice its bandwidth to allow exact recovery. In this case the band-limitedness of the signal is
1 the extra known information that allows us to reconstruct the signal from compressed measurements. In
.
1 compressedsensingtheadditionalstructureconsideredisthatthesignalissparsewithrespecttoacertain
0 knownbasis.AsopposedtosamplingatNyquistrateandsubsequentlycompressing,wenowobtainlinear
3
measurements of the signal and the compression and measurementsteps are now combined by obtaining
1
much smaller number of linear measurements than what would be in general required to reconstruct the
:
v signal from its measurements.
i
X After fixing the basis with respect to which the signal is sparse, the process of obtaining the measure-
mentscanbewrittenasy=Ax,where,y m isthevectorofmeasurements,x n isthesignaland
r
a A m n representsthemlinearfunction∈alRsactingonthesignalx.WecallAthe∈mReasurementmatrix.
×
∈R
The signal x is considered to have at most k non-zero componentsand we are typically interested in the
scenario where k is much smaller than n. Compressed Sensing revolves around the fact that for sparse
signals, the number of such linear measurements needed to reconstruct the signal can be significantly
smaller than the ambientdimension of the signal itself. The reconstructionproblem can be formulatedas
finding the sparsest solution x satisfying the constraints imposed by the linear measurementsy. This can
LaboratoryofInformationandDecisionSystems,DepartmentofElectricalEngineeringandComputerScience,MIT,Cambridge,
MA,02139,[email protected], [email protected]
2
be represented by
min x
0
|| ||
subject to y=Ax.
This problem is inherently combinatorial in nature and is in general a NP-hard problem. This is because
foracertainvalueofthesize ofthesupportofx givenbyk, oneneedsto searchthroughall n possible
k
supportsofthesignal.SeminalworkbyCande´sandTaoin[1]andDonohoin[2]showthatundercertain
(cid:0) (cid:1)
conditionsonthemeasurementmatrixA,ℓ normminimization,whichcanberecastasa linearprogram,
1
can recover the signal from its measurements. Additionally, a random matrix with i.i.d. Gaussian entries
with mean zero satisfies the required condition with overwhelming probability. Linear programming is
knowntohavepolynomialtimecomplexityandtheabovementionedresulttellsusthatforalargeclassof
measurementmatrices A we can solve an otherwise NP-hard combinatorialproblem in polynomialtime.
Subsequently,iterativemethodsbasedonagreedyapproachwereformulatedwhichrecoversparsesignals
fromtheir measurementsby obtainingan increasinglyaccurateapproximationto the actualsignal in each
iteration.ExamplesoftheseincludeCoSaMP[3]andIHT[4].Thecompressedsensingframeworkhasalso
been generalized to the problem of recovering low rank matrices from compressed linear measurements
represented by linear matrix equations [5] [6].
MostoftheearlierliteratureonCompressedSensingfocusedonthecase wheretheonlyconstraintson
thesignalx arethoseimposedbyitsmeasurementsy. Ontheotherhanditisnaturalto considerthecase
where besides sparsity, there is certain additional information on the structure of the underlying signal.
Thisistrue in severalapplications,examplesofwhichincludeencodingnaturalimages(JPEG),MRI and
DNA microarrays[7], [8]. This leads us to Model Based Compressed Sensing where the aim is to devise
recoverymethodsspecific to the signal model at hand. Furthermore,one would also want to quantify the
possible benefits it has over the standard method (e.g. lesser number of required measurements for the
same level of sparsity of the signal). This has been explored in some recent papers. The authors in [9]
analyzed a deterministic signal model, were the supportof the underlyingsignal is constrained to belong
to a given known set. This defines a subset of the set of all k-sparse signals, which is now the set of
M
allowable signals. This results in an additional constraint on the original reconstruction problem.
min x
0
|| ||
subject to y=Ax,
x .
∈M
Itwasshownthatanintuitivemodificationtothe CoSaMPorIHTmethodsucceedsin suitablyexploiting
the information about the model. The key property defined in [1], known as the Restricted Isometry
Property was adapted in [9] to a model based setting. With this, it was shown that results similar to [1]
can be obtained for model-based signal recovery.
As opposed to this, a probabilistic model, i.e. a Bayesian setting, was considered in [10]. Under this
model there are certain known probabilities associated with the componentsof the signal x. Specifically,
p ,i=1,2,...,n with 0 p 1 are such that
i i
≤ ≤
P(x is non-zero)=p i=1,2,...,n.
i i
The deterministic version of the same model called the “nonuniform sparse model” was considered in
[11]. For this, the use of weighted ℓ -minimization was suggested, given by
1
min x
w,1
|| ||
subject to y=Ax,
3
where x = n w x denotes the weighted ℓ norm of x. The quantities w ,i =1,2,...,n are
|| ||w,1 i=1 i| i| 1 i
some positive scalars. Similar to [2], [12], ideasbased on high dimensionalpolytopegeometrywere used
P
to provide sufficient conditions under which weighted ℓ -minimization recovers the sparse signal. This
1
method was introduced earlier in [13] where it was used to analyze the robustness of ℓ -minimization
1
in recovering sparse signals from noisy measurements. The specific model considered in [10] can be
described as follows. Consider a partition of the indices 1 to n into two disjoint sets T and T . Let
1 2
p = P ,i T and p = P ,i T . As a natural choice choose the weights in the weighted ℓ -
i 1 1 i 2 2 1
∈ ∈
minimization as w = W ,i T and w = W ,i T . The main result of [10] is that under certain
i 1 1 i 2 2
conditionson W2, P2 andA,∈weightedℓ -minimizati∈oncanrecoverasignaldrawnfromtheabovemodel
W1 P1 1
with overwhelming probability.
In this paper, we extend this approach to a more general class of signal models. In our model, the
probabilities p as described above are assumed to vary in a “continuous” manner across the indices.
i
The specific way in which the probabilities vary is governed by a known shape function p(.) which is
continuous, non-negativeand non-increasing.For a signal drawn from this model, we propose the use of
weightedℓ -minimizationtoreconstructitfromitscompressedlinearmeasurements.Inaddition,tomatch
1
thesignalmodel,weproposethattheweightsbechosenaccordingtoacontinuous,non-negativeandnon-
decreasingshapefunctionf(.). We provethatundercertainconditionsonp(.), f(.)andthemeasurement
matrix A, we can reconstruct the signal perfectly with overwhelmingprobability. We also suggest a way
to select the shape function f(.) for the weights based on p(.). Our results are also applicable when the
functions p(.) and f(.) are piece-wise continuous and the results in [10] can be obtained from ours as a
special case.
Therestofthepaperisorganizedasfollows.InSectionII,weintroducethebasicnotation.Weformulate
the exact questions that we set out to answer in this paper and state our main theorem. In Section III we
focus on how weighted ℓ -minimization behaves by restricting our attention to special class of signals
1
that are particularly suitable for the ease of analysis. We later show that the methods generalize to other
simple classes and is all we need to establish the main result of this paper. In Section IV we describe the
key featuresof a typical signal drawnfrom our model. We also provea suitable large deviationresult for
the probability that a signal drawn from our model lies outside this typical set. In Section V we provide
numerical computations to demonstrate the results we derive. We also provide simulations to reinforce
our results. We then conclude the paper with possibly interesting questions that may be explored in the
future.
II. PROBLEM FORMULATION
A. Notation and parameters
We denote scalars by lower case letters (e.g. c), vectors by bold lower case letters (e.g. x), matrices
with bolduppercase letters(e.g.A). Probabilityofan eventE is denotedbyP(E). The ith standardunit
vector in n is denoted by e =(0,0,...,1,...,0)T, where the “1” is located in the ith position.
i
The unRderlying sparse signal is represented by x n, the measurement matrix by A m n.
×
∈ R ∈ R
The vector of observations is denoted by y and is obtained through linear measurements of x given by
y = Ax. Typically we would need n linear measurements to be able to recover the signal. The scalar
α = m determines how many measurements we have as a fraction of n. We call this the compression
n
ratio.
B. Model of the Sparse Signal
Letp:[0,1] [0,1]beacontinuousmonotonicallynon-increasingfunction.Wecallp(.)theprobability
→
shapefunctionandthereasonforthisnamewillbecomeclearfromthedescriptionbelow.Thesupportof
4
the signal x is decided by the outcome of n independentBernoulli random variables. In particular, if E
i
denotestheeventthati Supp(x),thentheeventsE i=1,...,nareindependentandP(E )=p i .
∈ i i n
Although we assume throughout the paper that p(.) is continuous, our result generalizes to piecewise
(cid:0) (cid:1)
continuous functions as well. Let us denote by k the cardinality of Supp(x). Note that under the above
model k is a random variable that can take any value from 0 to n. The expected value of k is given by
n p i . We denote by δ the expected fractional sparsity given by δ = 1E[k]= 1 n p i .
i=1 n n n i=1 n
As is standard in Compressed Sensing literature we assume that the entries of the measurementmatrix
P (cid:0) (cid:1) P (cid:0) (cid:1)
A are i.i.d.Gaussian randomvariableswith meanzero and varianceone. The measurementsare obtained
as y=Ax.
C. Weighted ℓ -minimization
1
When the fractional sparsity δ is much smaller than one, a signal sampled from the model described
above is sparse. Hence, it is possible to recover the signal from its measurements y by ℓ -minimization
1
which is formulated as
min x
1
|| ||
subject to Ax=y.
Howeverthisdoesnotexploittheextrainformationavailablefromtheknowledgeofthepriors.Instead,we
useweightedℓ -minimizationtorecoverthesparsesignalwhichiscapturedbythefollowingoptimization
1
problem:
min x w,1 (1)
|| ||
subject to Ax=y.
wherew ∈Rn is a vectorof positiveweightsand ||x||w,1 = ni=1wi|xi| refersto the weightedℓ1 norm
of x, for a given weight vector w. The weight vector w plays a central role in determining whether (1)
P
successfully recovers the sparse signal x. Intuitively, w should be chosen in a certain way depending on
p(.)soastoobtainthebestperformance(althoughatthispointwehavenotpreciselydefinedthemeaning
of this). Keeping in mind the structure of p , i = 1,...,n, we suggest using weights w , i = 1,...,n
i i
whichhaveasimilarstructure.Formally,letf :[0,1] + beanon-negativenon-decreasingcontinuous
function. Then we choose the weights as w =f i →. WRe call f(.) the weights shape function.
i n
(cid:0) (cid:1)
D. Problem statement
In this paper we try to answer the following two questions:
Giventhe problemparameters(size of the matrix definedby m,n),the functionsp(.) and f(.), does
•
weighted ℓ -minimization in (1) recover the underlying sparse signal x with high probability?
1
Given a probability shape function p(.), and a family of weight shape functions , how to choose
• W
a function f(.) that has the best performance guarantees?
∈W
We give an answer in the affirmative to the first question, given that the functionsp(.) and f(.) satisfy
certain specified conditions. This is contained in the main result of this paper which is
Theorem 1. Let the probability shape function p(.) and the weight shape function f(.) be given. Let E
be the event that weighted ℓ -minimization described in (1) fails to recover the correct sparse vector x.
1
There exists aquantityψ¯ (p,f)whichcanbecomputedexplicitly asdescribedinSectionB-4, suchthat
tot
wheneverψ¯ (p,f)<0theprobabilityoffailureP(E)ofweightedℓ -minimizationdecaysexponentially
tot 1
5
with respectto n. More precisely, if forsome ǫ>0 we have ψ (p,f) ǫ. then there exists a constant
tot
≤−
c(ǫ)>0 such that for large enough n, the probability of failure satisfies P(E) e nc(ǫ).
−
≤
Toanswerthesecondquestion,wefirstdefinethemeasureofperformance.Foragivenvalueofα= m,
n
let δ¯be the maximum value of δ for which weighted ℓ -minimization can be guaranteedto succeed with
1
highprobability.Foragivenfamilyofweightfunctions ,theweightfunctionf(.)thathasthehighestδ¯
W
is said to have the best performance.In thispaper we will describe a methodto choose a weightfunction
in and demonstrate the method in Section V for a linearly parameterized family.
W
III. ANALYSIS OF WEIGHTEDℓ1-MINIMIZATION
Inthissection,weanalyzetheperformanceofweightedℓ -minimizationwithweightsspecifiedbyf(.)
1
on a certain special class of sparse signals x. As we will see in Section IV, we can easily generalize the
analysis to the signals drawn from our signal model.
Recall that the failure event E from Theorem 1 is defined as the event that weighted ℓ -minimization
1
failstorecoverthecorrectsparsevectorx.WecalltheprobabilityofthiseventP(E)astheprobabilityof
failure. Without loss of generality we can assume that x w,1 =1, that is x lies on a k 1-dimensional
|| || −
face of the weighted cross-polytope
, x n s.t. x w,1 1
P { ∈R || || ≤ }
which is the weighted ℓ -ball in n dimensions. The specific face of on which x lies is determined by
1
P
the support of x. The probability of failure P(E) can be written as
P(E)= P(x F)P(E x F),
∈ | ∈
F
X∈F
where is the set of all faces of the polytope . Then, as shown in [10], the event E x F is
F P { | ∈ }
preciselythe eventthatthereexistsa non-zerou null(A)suchthat x+u w,1 x w,1 conditioned
∈ || || ≤|| ||
on x F . Also since A has i.i.d. Gaussian entries, sampling from the nullspace of A is equivalentto
{ ∈ }
sampling a subspace uniformly from the Grassmann manifold Gr (n). So conditioned on x F
(n m)
the event E x F is same as the eventthat a uniformlychosen (n− m)-dimensionalsubspac{e sh∈ifte}d
{ | ∈ } −
to x intersects the polytope at a point other than x. The probability of the above event is also called
P
the complementaryGrassmannAngleforthe face F with respectto the polytope undertheGrassmann
P
manifoldGr (n).BasedonworkbySantalo´ [14]andMcMullen[15]theComplementaryGrassmann
(n m)
Angle can be e−xpressed explicitly as the sum of products of internal and external angles.
P(E x F)=2 β(F,G)γ(G, ) (2)
| ∈ P
Xs≥0G∈J(mX+1+2s)
where β(F,G) and γ(G, ) are the internal and external angles and J(r) is the set of all r-dimensional
P
faces of . The definitions of internal and external angles can be found in [10]). We include them here
P
for completeness.
Theinternalangleβ(F,G)isthefractionofthevolumeoftheunitballcoveredbytheconeobtained
•
by observing the face G from any any point in face F. The quantity β(F,G) is defined to be zero
if F is not contained in G and is defined to be one if F =G.
The external angle γ(G, ) is defined to be the fraction of the volume of the unit ball covered by
• P
the cone formed by the outward normals to the hyperplanes supporting at the face G. If G=
P P
then γ(G, ) is defined to be one.
P
6
In this section we describe a method to obtain upperboundsfor P(E x F) by finding upperbounds
| ∈
on the internal and external angles described above. We first analyze P(E x F) for the “simplest”
| ∈
class of faces. We denote by Fk, the face whose vertices are given by 1 e , 1 e ,..., 1 e . Thus the
0 w1 1 w2 2 wk k
vertices are defined by the first k indices and we call this face the “leading” k 1-dimensional face of
−
. We will spend much of this section developing bounds for P(E x Fk) for such “leading” faces.
P | ∈ 0
Then in Section IV, we describe the typical set of our signal model and show that for the purposes of
boundingP(E) it is sufficient to consider a certain special class of faces. The boundswe develop in this
section for “leading” faces can be easily generalized to faces belonging to this special class.
A. Angle Exponents for leading faces
Wedefinethefamilyofleadingfaces asthesetofallfaceswhoseverticesaregivenby 1 e , 1 e ,..., 1 e
F1 w1 1 w2 2 wk k
for some k. In this section we will establish the following result which bounds the internal and external
angles related to a leading face of . We will then use this result in the next section to providean upper
P
bound on P(E x F) for F .
1
| ∈ ∈F
Theorem 2. Let k = δ and l = τ with τ > δ. Let Gl be the face whose vertices are given by
n n 0
1 e , 1 e ,..., 1 e and let Fk Gl be the face whose vertices are given by 1 e , 1 e ,..., 1 e .
Twh1en1 twh2ere2 existwqlualntities ψ¯ 0(τ⊂) an0d ψ¯ (δ,τ) which can be computed explwic1itl1y wa2s d2escribwekd ikn
ext int
Section III-C2 and Section III-C3 respectively, such that for any ǫ > 0, there exist integers n (ǫ) and
0
n (ǫ) satisfying
1
n 1log(β(Fk,Gl))<ψ¯ (δ,τ)+ǫ, for all n>n (ǫ),
• n−1log(γ(G0l, )0)<ψ¯ in(tτ)+ǫ, for all n>n (ǫ)0.
• − 0 P ext 1
The quantities ψ¯ (δ,τ) and ψ¯ (τ) are called the optimized internal angle exponent and the external
int ext
angle exponent respectively.
Wheneveranyoftheangleexponentsdescribedaboveisnegative,thecorrespondingangledecaysatan
exponentialratewithrespecttotheambientdimensionn.WewillproveTheorem2overthefollowingtwo
subsections by finding the optimized internal and external angle exponents that satisfy the requirements
of the theorem.
1) InternalAngleExponent: Inthissubsectionwefindtheoptimizedinternalangleexponentψ¯ (δ,τ)
int
that satisfies the conditions described in Theorem 2. We begin by stating the following result from [10]
which providesthe expression for the internal angle of a face F with respect to a face G in terms of the
weights associated with their vertices. Subsequently,we find an asymptotic upper bound on the exponent
of the internal angle using that expression. In what follows, we denote by HN(0,σ2) the distribution of
a half normal random variable obtained by taking the absolute value of a N(0,σ2) distributed random
variable.
Lemma 1. [10] Define σ = j w2. Let Y N(0,1) be a normal random variable and Y
i,j p=i p 0 ∼ 2 p ∼
w2
HN(0,2σp1+,kk) for p = 1,2,...,lP−k be independent half normal distributed random variables that are
independent of Y0. Define the random variable Z ,Y0− pl−=k1Yp. Then,
β(Fk,Gl)= √π σP1,lp (0),
0 0 2l k σ Z
− r 1,k
where p (.) is the density function of Z.
Z
7
We now proceed to derive an upper bound for the quantity p (0). Much of the analysis is along the
Z
lines of [2]. Let the random variable S be defined as S = pl−=k1Yp. So, Z = Y0 −S and using the
convolution integral, the density function of Z can be written as
P
∞ ∞
p (0)= p ( v)p (v)dv =2 vp (v)F (v)dv,
Z Y0 − S Y0 S
Z−∞ Z0
where F (v) is the cumulative distribution function of S. Let µ be the mean of the random variable S.
S S
Then,
µS ∞
p (0)=2 vp (v)F (v)dv+2 vp (v)F (v)dv.
Z Y0 S Y0 S
Z0 ZµS
I II
As in [2], the second term satis|fies II <{zµ∞S2vpY0(v})dv|= e−µ2S.{Azs we will}see later in this section
µ2 cn for some c > 0 and hence II e c2n2. Since we are interested in computing the asymptotic
S ∼ ∼R−
exponentofp (0), we can ignoreII fromthiscomputation.To boundI, we use F (v) exp( λ (v)),
Z S ≤ − ∗S
where λ (v) denotes the rate function (convex conjugate of the characteristic function λ (.)) of the
∗S S
random variable S. So we get
I 2 µSve−v2−λ∗S(v)dv. (3)
≤ √π
Z0
For ease of notation we define the following quantities.
l
s , w
k+1,l p
p=k+1
X
1
λ(s), s2+log(2Φ(s))
2
l k
λ (s), 1 − λ(w s)
0 p+k
s
k+1,l
p=1
X
λ∗0(y),msaxsy−λ0(s).
Here λ(s) is the characteristic function of the standard half normal random variable and Φ(x) is the
standard normal distribution function. Using the above definition, we can express the relation between
λ (y) and λ (y) as
∗S ∗0
1 s
k+1,l
λ (y)= λ y
∗0 sk+1,l ∗S(cid:18)√σ1k (cid:19)
We compute
l k
µ = − EY = 2 sk+1,l
S p
p=1 rπ√σ1,k
X
Changing variables in (3) by substituting v = sk+1,ly, we get
√σ1,k
I 2 s2k+1,l √π2 y exp[ s2k+1,ly2 s λ (y)]dy.
≤ √π 2σ −2σ − k+1,l ∗0
1,k Z0 1,k
8
Now, as w =f i , we have
i n
(cid:0) (cid:1) l i τ
s = f =n f(x)dx+o(1) .
k+1,l
n
i=k+1 (cid:18) (cid:19) (cid:18)Zδ (cid:19)
X
Define c (δ,τ), τ f(x)dx. This gives us s =n(c (δ,τ)+o(1)). Similarly,
0 δ k+1,l 0
R k k i δ
σ = w2 = f2 =n( f2(x)dx+o(1))=n(c (δ)+o(1)),
1,k i n 1
i=1 i=1 (cid:18) (cid:19) Z0
X X
where c (δ) is defined as c (δ), δf2(x)dx. This gives us
1 1 0
s2 R c2
2σk+1,l y2+sk+1,lλ∗0(y)=n 2c0 y2+c0λ∗(y)+o(1) =n(η(y)+o(1)).
k+1,l (cid:18) 1 (cid:19)
where η(y) is defined as η(y) , 2cc201y2+c0λ∗0(y) . Using Laplace’s method to bound I as in [2], we
get
(cid:16) (cid:17)
I Rne−nη(y∗), (4)
≤
where n 1log(R )=o(1), and y is the minimizer of the convex function η(y). Let
− n ∗
λ (y )=max sy λ (s)=s y λ (s ).
∗0 ∗ s ∗− 0 ∗ ∗− 0 ∗
′ ′
Then the maximizing s satisfies λ (s ) = y . From convex duality, we have λ (y ) = s . The
∗ 0 ∗ ∗ ∗0 ∗ ∗
minimizing y of η(y) satisfies
∗
c20y +c λ ′(y )=0
c ∗ 0 ∗0 ∗
1
c2
= 0y +c s =0. (5)
∗ 0 ∗
⇒ c
1
This gives
λ′(s )= c1s . (6)
0 ∗ −c ∗
0
First we approximate λ (s) as follows
0
1 l−k 1 l p 1 τ
λ (s)= λ(w s)= λ f s +o(1)= λ(sf(x))dx+o(1).
0 p+k
s nc n c
k+1,l Xp=1 0 p=Xk+1 (cid:16) (cid:16) (cid:17) (cid:17) 0 Zδ
From the above we obtain
d 1 τ ′
λ (s)= f(x)λ (sf(x))dx+o(1).
0
ds c
0 Zδ
Combining this with equation (6), we can determine s up to a o(1) error by finding the solution to the
∗
equation
τ
′
f(x)λ (s f(x))dx+c s =0. (7)
∗ 1 ∗
Zδ
9
We define the internal angle exponent as
ψ (β,τ,y), (τ δ)log2 η(y) (8)
int
− − −
c2
= (τ δ)log2 0 y2+c λ (y) , (9)
− − − 2c 0 ∗0
(cid:18) 1 (cid:19)
and the optimized internal angle exponent as
ψ¯ (β,τ),maxψ (β,τ,y)= (τ δ)log2 η(y ) (10)
int int ∗
y − − −
c2
= (τ δ)log2 0 y 2+c λ (y ) , (11)
− − − 2c ∗ 0 ∗0 ∗
(cid:18) 1 (cid:19)
where y is determined through s from equation (5) and s is determined by solving the equation (7).
∗ ∗ ∗
From inequality (4) and Lemma 1 we see that the function ψ¯ (β,τ) satisfies the conditions described
int
in the first part of Theorem 2.
2) ExternalAngleExponent: Inthissubsectionwefindtheoptimizedexternalangleexponentψ¯ (τ)
ext
thatsatisfies the statementof Theorem2. We proceedsimilar to the previoussection and beginby stating
the following lemma from [10] which provides the expression for the external angle of a face G with
respect to the polytope in terms of the weights associated with its vertices. We then find an upper
P
bound on the asymptotic exponent of the external angle.
Lemma 2. [10] The external angle γ(Gl, ) is given by
0 P
n wpx
γ(Gl0,P)=π−n−2l+12n−lZ0∞e−x2p=l+1 Z0√σ1,l e−yp2dyp!dx.
Y
To simplify the expression for the external angle, we define the standard error function as
2 x
erf(x)= e t2dt
−
√π
Z0
and rewrite the external angle as
n
γ(Gl0,P)= σπ1,l ∞e−σ1,lx2 erf(wix)dx.
r Z0 i=l+1
Y
Similar to the method used in the previous section
l τ
σ = w2 =n f2(x)dx+o(1) =nc (τ)+o(1).
1,l i 2
i=1 (cid:18)Z0 (cid:19)
X
where c (τ) is defined as c (τ), τf2(x)dx. Substituting this we have
2 2 0
γ(Gl, )= σ1,lR ∞exp n c x2 1 n log(erf(w x)) dx
0 P r π Z0 "− 2 − ni=l+1 i !#
X
σ1,l ∞
= exp[ nζ(x)]dx.
π −
r Z0
10
where ζ(x) is defined as ζ(x) , c x2 1 n log(erf(w x)) . Again using Laplace’s method we
2 − n i=l+1 i
get
(cid:0) P (cid:1)
γ(Gl, ) R exp[ nζ(x )], (12)
0 P ≤ n − ∗
wherex istheminimizerofζ(x)andn 1log(R )=o(1).Theminimizingx satisfies2c x =G′(x ),
∗ − n ∗ 2 ∗ 0 ∗
where
n
1
G (x), log(erf(w x)).
0 i
n
i=l+1
X
We first approximate G (x) as follows:
0
1 n 1 n 1
G (x)= log(erf(w x))= log(erf(f(i/n)x))= log(erf(xf(y)))dy+o(1).
0 i
n n
i=l+1 i=l+1 Zτ
X X
So the minimizing x can be computed up to an error of o(1) by solving the equation
∗
1 f(y)erf′(x(f(y))
2c x = dy. (13)
2 ∗
erf(xf(y))
Zτ
We define the external angle exponent as
ψ¯ (τ,x), ζ(x) (14)
ext
−
and the optimized external angle exponent as
1
ψ¯ (τ),maxψ¯ (τ)= ζ(x )= c x 2 log(erf(x f(y)))dy . (15)
ext ext ∗ 2 ∗ ∗
x − − −
(cid:18) Zτ (cid:19)
where x can be obtained by solving equation (13).
∗
From (12) it is clear that this function satisfies the conditions in the second part of Theorem 2. This
completes the proof of both parts of the theorem.
B. Recovery Threshold for the family of leading faces
In this section we use the bounds on the asymptotic exponents of the internal and external angle from
Theorem 2 to find an upper bound for P(E x F) for F . The main result of this section is the
1
| ∈ ∈ F
following theorem.
Theorem 3. Let F =Fk for k =δn. F be the face with k =δn. There exists a function ψ¯ (δ)
0 δ ∈F1 tot
whichwe callthetotalexponentofFk suchthatgivenǫ>0, thereexists n(ǫ)suchthatforalln>n(ǫ),
0
1 log(P(E x Fk))< nψ¯ (δ)+ǫ. The function ψ¯ (δ) can be computed explicitly as described in
n | ∈ 0 − tot tot
(16).
Using the decomposition equation (2) and the fact that β(F,G) is non-zero only if F G, we get
⊆
P(E x F) 2 β(F,G)γ(G, ).
| ∈ ≤ P
(l>m)(G F, G J(l))
X ⊇ X∈
Recall that J(r) is the set of all r-dimensional faces of .
P
To proceed, we will need the following useful lemma (proof can be found in the appendix).