The smallest singular value of deformed random rectangular matrices
Fan Yang
∗11
Department of Mathematics, University of Wisconsin-Madison February 8, 2017
Abstract
We prove an estimate on the smallest singular value of a type of multiplicatively and addi- tively deformed random rectangular matrices. Suppose nďN ďM ďCN for some positive constantC. LetXbe a randomMˆnmatrix with independent entries, which have zero mean, unit variance and arbitrarily high moments. Let T be an NˆM deterministic matrix with comparable singular valuescďsNpTq ďs1pTq ďC, and letAbe anNˆndeterministic matrix with}A} “Op?
Nq. Then we prove that with high probability, the smallest singular value of T X´A is at least of the order N´p?
N´?
n´1q for anyą0. If we assume further the entries ofX have subgaussian decay, then the smallest singular value ofT X´A is at least of the order?
N´?
n´1 with high probability, which is an essentially optimal bound.
1 Introduction
1.1 Smallest singular values of random matrices
In this paper, we prove a lower bound for the smallest singular values of deformed random rectangular matrices of the form T X ´A, where T, A are fixed matrices and X is a random matrix with independent centered entries.
Consider anN ˆnreal or complex matrix A with N ěn. The singular values sipAqof Aare the eigenvalues of pA˚Aq1{2 arranged in the non-increasing order
s1pAq ěs2pAq ě. . .ěsnpAq.
Of particular importance are the largest singular values1pAq, which controls the norm}A}, and the smallest singular valuesnpAq, which measures the invertibility ofA˚A.
A natural random matrix model is given by a rectangular matrixXwhose entries are independent centered random variables with unit variance and certain moment assumptions. In this paper, we focus on random variables witharbitrarily high moments(see (1.8)). This includes all thesubgaussian and subexponential variables, and some other random variables with heavy tails. The asymptotic behavior of the extreme singular values of X has been well-studied. Let X be an Nˆn random
∗E-mail: fyang75@math.wisc.edu.
matrix. Let λ1ěλ2ě. . . ěλn be the eigenvalues ofN´1X˚X and define the empirical spectral distribution as µN :“ n´1řn
i“1δλi. If n{N Ñ λ P p0,1q as N Ñ 8, then µN converges (in distribution) to the famous Marchˇenko-Pastur (MP) law almost surely [13]. Moreover, the MP distribution has a density with positive support on rp1´
?λq2,p1`
?λq2s, which suggests that
asymptotically, s1pXq Ñ
? Np1`
? λq “?
N`?
n, and snpXq Ñ? Np1´
? λq “?
N´?
n. (1.1) The almost sure convergence of the largest singular value was proved in [7] for random matrices whose entries have arbitrarily high moments. The almost sure convergence of the smallest singular value was proved in [20] for Gaussian random matrices (i.e. the Wishart matrix). These results were later generalized to random matrices with i.i.d entries with finite fourth moment in [24] and [3].
A considerably harder problem is to establish non-asymptotic versions of (1.1), which would hold for any fixed dimensionsN andn. Most often needed are upper bounds for the largest singular value s1pXqand lower bounds for the smallest singular value snpXq. With a standard -net argument, it is not hard to show that the operator norm of X is at most of the optimal order ?
N for all dimensions, see e.g. [6, 11, 17]. On the other hand, the lower bound for the smallest singular value is much harder to obtain. There has been much progress in this direction during last decade.
Tall matrices. It was proved in [12] that for arbitrary aspect ratiosλă1´c{logN and random matrices with independent subgaussian entries, one has
P
´
snpXq ďcλ
? N
¯
ďe´cN, (1.2)
where cλą0 depends only onλand the maximal subgaussian moment of the entries.
Square matrices. For square random matrices (i.e. whenN“n), the estimate on the smallest singular value was first obtained in [15], where it was proved that for subgaussian matrixX,snpXq ě N´3{2 with high probability. This result was later improved to [16]
P
´
sNpXq ďN´1{2¯
ďC`e´cN, (1.3)
and essentially optimal result for subgaussian matrices. Subsequently, the lower bound for sNpXq was proved under weakened moments assumptions [21, 14, 9]. In [22], Tao and Vu provedsNpXq ě N´Op1q with probability 1´op1qassuming only unit variance of matrix entries.
Almost square matrices. The gap 1´c{logN ďλă1 was filled in [17]. It was shown that for subgaussian random rectangular matrices,
P
´
snpXq ďp? N´?
n´1q
¯
ď pCqN´n`1`e´cN, (1.4) for all fixed dimensionsN ěn. This bound is essentially optimal for subgaussian matrices with all aspect ratios. It is easy to see that (1.2) and (1.3) are now the special cases of (1.4).
There is an alternative way to study the extreme singular values, that is, through a local estimate (often referred to aslocal law) on the Stieltjes transform of the empirical spectral distributionµN. Following this approach, one can prove the following finer results forX whose entries have arbitrarily high moments [5]: for any (small) ą0 and (large)Dą0, there existsnp, Dqsuch that
P
´ˇ ˇ
ˇs1pXq ´´?
N`? n
¯ˇ ˇ
ˇě pN nq1{4n´2{3`
¯
ďn´D, for allněnp, Dq. (1.5) If in addition |λ´1| ěcfor some constantcą0 (so that the left edgep1´?
λq2 of the MP law is strictly away from zero), then a similar result holds for the smallest singular value:
P
´ˇ ˇ
ˇsnpXq ´
´?
N´? n
¯ˇ ˇ
ˇě pN nq1{4n´2{3`
¯
ďN´D, for allněnp, Dq. (1.6)
Here the assumption |λ´1| ěc is crucial. The bound (1.6) does not hold forλÑ1 case.
In this paper, we are interested in the extreme eigenvalues of a multiplicatively and additively deformed random rectangular matrix. Given anMˆnrandom matrixX with independent entries, we consider the matrix T X ´A, where T and A are deterministic N ˆM and N ˆn matrices, respectively. It is easy to control the largest singular value through}T X´A} ď }T}}X} ` }A}. On the other hand, we expect that if nď N ďM and the singular values of T satisfy c ďsNpTq ď s1pTq ďC, then the boundsnpT X´Aq ďp?
N´?
n´1qwould hold for subgaussianX. Intuitively, we can regardT as a nice “isomorphic projection” ofRM onto anN-dimensional subspace. On the other hand, by (1.4), X is a nice “isomorphic embedding” of Rn into RM with high probability.
Hence T X should be a nice embedding of Rn into an N-dimensional space. Moreover, adding a fixed matrix A can make the matrix become singular with very small probability (actually with 0 probability if the entries of X have continuous distributions). ThusT X´Ashould have a suitable lower bound for the smallest singular value. In this paper, we will make this argument rigorous.
One of our motivations is the potential application in statistical science. Consider sample covari- ance matrices of the formQ“n´1BB˚, whereBis anNˆnmatrix. The columns ofB representn independent observations of some randomN-dimensional vectorb. For the sample vectorb, we take a linear modelb“Tx, whereT is a deterministicNˆM matrix andxis a randomM-dimensional vector with independent entries. Then we can writeb“Tˆx`a, where ˆxis a centered random vector anda“TEx. In addition, we may without loss of generality assume that the entries of ˆxhave unit variance by absorbing the variance of ˆxi intoT. Hence we can writeB into the formB“T X´A, and our result provides a good a priori estimate on the smallest singular values of B. It is worth mentioning that in [10], the authors proved a local law for sample covariance matrices of the form Q“n´1T XX˚T˚. As a corollary, they showed that there exist constants 0ăa1pλ, Tq ăa2pλ, Tq, depending only on λ and the singular values of T, such that (1.5) and (1.6) hold if one replace
?N`?
nwith?
N a2pλ, Tqand? N´?
nwith?
N a1pλ, Tq. However, on the one hand, the local law forn´1pT X´AqpT X´Aq˚ is still unknown; on the other hand, (1.6) fails whena1pλ, Tq Ñ0 (which may happen, for example, when λÑ1).
Another application of our result is the circular law for square random matrices. Let X be an N ˆN random matrix with i.i.d. entries with mean zero and variance one. It is well known that the spectral measure of eigenvalues of N´1{2X converges to the circular law, i.e. the uniform distribution on the unit disk [8, 1]. An important input of the proof is the upper and lower bounds for the extreme singular values ofN´1{2X´zfor any fixedzPC[9, 14, 21, 22]. In [23], we obtained a generalizedlocal circular lawfor square random matrices of the formN´1{2T X. In order to get a lower bound for the smallest singular value ofN´1{2T X´z, we originally assumed that the entries of X have continuous distributions. This assumption rules out some important models such as the Bernoulli random matrices. Now with the result of this paper, we can relax this assumption greatly to include all random variables with sufficiently high moments.
1.2 Main results and reduction to subgaussian matrices
Letξ1, . . . , ξn be independent random variables such that for all 1ďiďn,
Eξi“0, E|ξi|2“1, (1.7)
and for any pPN, there is anN-independent constantCpsuch that
E|ξi|pďCp. (1.8)
We assume that X is an Mˆnrandom matrix, whose rows are independent copies of the random vectorpξ1, . . . , ξnq. In this paper, we want to prove a lower bound for the smallest singular value of
T X´B, where T andB are fixedNˆM and Nˆnmatrices, respectively. We assume that nďN ďM ďCN, }B} ďC?
N (1.9)
for some constantCą0. Moreover, we assume the eigenvalues ofT T˚ satisfy that
Cěσ1ěσ2ě ¨ ¨ ¨ ěσN ěC´1. (1.10) For definiteness, in this paper we focus onreal matrices. However, our results and proofs also hold, after minor changes, in the complex case if we assume in addition that Xij have independent real and imaginary parts, such that
EpReXijq “0, EpReXijq2“1 2,
and similarly for ImXij. The main result of this paper is the following theorem.
Theorem 1.1. Suppose the assumptions (1.7), (1.8), (1.9) and (1.10) hold. Then for everyτ ą0, ě0 andDą0, there exists a constant Cą0 such that
P
´
snpT X´Bq ďN´τp? N´
?n´1q
¯
ď pCqN´n`1`N´D (1.11) for large enough N.
To prove this theorem, we first truncate the entries of X at level Nω for some small ω ą 0.
Combining condition (1.8) with Markov’s inequality, we get that for any (small) ωą0 and (large) Dą0, there existsNpω, Dqsuch that
Pp|ξi| ąNω{2q ďN´D´2
for allN ěNp, Dq. Hence with a loss of probabilityOpN´Dq, it is sufficient to control the smallest singular values of the random matrix TX˜´B, where
X˜ “1ΩX, Ω :“ t|Xij| ďNω{2 for all 1ďiďM,1ďj ďnu.
By (1.8) and integration by parts, we can verify that E`
ξi1t|ξi|ďNω{2u
˘“O`
N´D´2`ω˘ , Var`
ξi1t|ξi|ďNω{2u
˘“1`O`
N´D´2`2ω˘
, 1ďiďn. (1.12) LetD1be annˆndiagonal matrix such that pD1qii “Var`
ξi1t|ξi|ďNω{2u
˘1{2
. LetT “UDV˜ be a singular decomposition, whereU is an NˆN unitary matrix,V is anMˆM unitary matrix and D˜ “ pD,0qsuch that D“diagpd1, d2, . . . , dNqwith |di|2“σi (so D is nonsingular by (1.10)). We denote V “
ˆ V1
V2
˙
, whereV1 has sizeNˆM andV2 has sizepM ´Nq ˆM. Then we have TX˜´B “U DV1pX˜´EX˜q´pB´TEX˜q “U D
”
V1pX˜ ´EXqD˜ 1´1´
´
D´1U´1B´V1EX˜
¯ D1´1
ı D1.
Due to (1.10) and (1.12), we only need to control snpV1Y ´Aq, where Y :“ pX˜ ´EX˜qD´11 and A:“ pD´1U´1B´V1EXqD˜ ´11 . Now by (1.9), (1.10), (1.12) and the definition of Ω, it is easy to see that Y is a random matrix with independent centered entries satisfying
EYij“0, VarpYijq “1, |Yij| ďNω, (1.13)
and Ais a deterministic matrix such that }A} ďC
´
}B} ` }EX}˜
¯
ďC´?
N`N´D´1`ω
¯ ďC?
N . (1.14)
Moreover, by Theorem 2.10 of [5], there exists a constant Cą0 such that Pp}X˚X} ďCNq ěN´D
for large enoughN. Then using}X˜} ď }X}, we get Pp}Y} ďC?
Nq ě1´N´D. (1.15)
Now it is easy to see that Theorem 1.1 is a simple corollary of the following theorem.
Theorem 1.2. Letξ1, . . . , ξnbe independent centered random variables with variance 1, finite fourth moment and subgaussian moment bounded by K for some K ”KpNq ďNω. Let Y be anM ˆn random matrix such that its rows are independent copies of the random vector pξ1, . . . , ξnq, and suppose that}Y} ďC?
N. LetP be anNˆM matrix withP PT “1, andAbe a fixedNˆnmatrix with }A} ďC?
N. Then for every ě0, there exists constantsc, Cą0 such that P
´
snpP Y ´Aq ď´?
N´?
n´1¯¯
ď`
CK12˘N´n`1
`e´cN{K4 (1.16) for large enough N.
Recall that a random variableξis called subgaussian if there existsKą0 such that
Pp|ξ| ątq ď2 expp´t2{K2q for alltą0. (1.17) The infimum of suchKis called the subgaussian moment ofξor theψ2-norm}ξ}ψ2. By (1.13),Yij
are uniformly bounded byNω, so they are subgaussian with}Yij}ψ2 ďNω. Combining with (1.15), we see that Y satisfies the assumptions in Theorem 1.2 with a loss of probabilityN´D. Note that the condition (1.17) can be equivalently formulated as the moment condition
pE|ξ|pq1{p ďCK?
p, for allpě1, (1.18)
which is stronger than our condition (1.8).
Remark 1.3. SupposeXij are subgaussian random variables with subgaussian moment bounded by a constant Ką0. Then
Pp}X} ětN1{2q ďe´c0t2N fortěC0,
wherec0, C0ą0depend only on the subgaussian momentK (see [17, Proposition 2.4]). Combining with Theorem 1.2, we obtain the following optimal bound
P
´
snpT X´Bq ď´?
N´
?n´1
¯¯
ď pCqN´n`1`e´cN (1.19) for subgaussian random matrices, which corresponds to the bound in (1.4).
Remark 1.4. TheK4 andK12 factors in Theorem 1.2 can be improved during the proof. However, we do not attempt to get the optimal factors in this paper.
The bulk of this paper is devoted to the proof of Theorem 1.2. In the preliminary Section 2, we introduce some notations and tools that will be used in the proof. In Section 3, we first reduce the problem into bounding below }pP Y ´Aqx}2 forcompressible unit vectorsxPSn´1, whose l2-norm is concentrated in a small number of coordinates, and forincompressible unit vectorscomprising the rest of the sphereSn´1. Then we prove a lower bound for compressible vectors using a large deviation estimate (Lemma 2.7) and a standard -net argument. The incompressible vectors are dealt with in Sections 4 and 5. In Section 4, we consider the case 1ďnăλ0N for some constant λ0 P p0,1q, i.e. when P Y ´A is a tall matrix. The proof can be finished with a small ball probability result (Lemma 2.6) and the-net argument. The almost square case with λ0N ďnďN is considered in Section 5. We first reduce the problem into bounding the distance between a random vector and a random subspace, and then complete the proof with the random distance Lemma 5.4, whose proof will be given in Section 6.
Acknowledgements. I would like to thank Roman Vershynin for valuable suggestions and for pointing out several useful references. I am also grateful to my advisor Jun Yin for his direction and helpful comments on this paper. Finally, many thanks to my friend Haokai Xi for useful discussions.
2 Basic notations and tools
In this paper, we use C to denote a generic large positive constant, which may depend on fixed parameters and whose value may change from one line to the next. Similarly, we use c, or ω to denote a generic small positive constant. If a constant depends on a quantitya, we useCpaqor Ca
to indicate this dependence.
The canonical inner product onRn is denoted x¨,¨y, and the Euclidean norm is denoted} ¨ }2. The distance from a point x to a setD in Rn is denoted distpx, Dq. The unit sphere centered at the origin in Rn is denoted Sn´1. The orthogonal projection inRn onto a subspace E is denoted PE. For a subset of coordinatesJ Ď t1, . . . , nu, we often writePJ forPRJ. The unit sphere ofE is denoted SpEq:“Sn´1XE.
For any (complex) matrix A, we use A˚ to denote its conjugate transpose, AT the transpose, }A}the operator norm and}A}HS the Hilbert-Schmidt norm. We usually write an identity matrix as 1 without causing any confusions.
The following tensorization lemma is Lemma 2.2 of [16]
Lemma 2.1 (Tensorization). Let ζ1, . . . , ζn be independent non-negative random variables, and let B, 0ě0.
(1) Assume that for each k,
Ppζkăq ďB for allě0. Then
P
˜ n ÿ
k“1
ζk2ă2n
¸
ď pCBqn for allě0, whereC is a universal constant.
(2) Assume that there exist λą0 andµP p0,1qsuch that for eachk, Ppζk ăλq ďµ.
Then there existsλ1ą0 andµ1P p0,1q that depend onλandµ only and such that P
˜ n ÿ
k“1
ζk2ăλ1n
¸ ďµn1.
Consider a subset D Ă Rn and ą0. An -net of D is a subset N ĂD such that for every x P D one has distpx,Nq ď . The following lemma about the cardinality of -nets is stated as Propositions 2.1 and 2.2 in [17].
Lemma 2.2 (Nets). Fix anyą0.
(1) There exists an -net of Sn´1 of cardinality at most 2n`
1`2´1˘n´1
.
(2) LetS be a subset ofSn´1. There exists an-net ofS of cardinality at most2n`
1`4´1˘n´1
. Next we define the small ball probability for a random vector.
Definition 2.3. The L´evy concentration function of a random vector S PRm is defined forą0 as
LpS, q “ sup
vPRm
Pp}S´v}2ďq,
which measures the small ball probabilities.
With Definition 2.3, it is easy to prove the following lemma. It will allow us to select a nice subset of the coefficients ak when computing the small ball probability.
Lemma 2.4. For anyσĎ t1, . . . , nu, any aPRn and any ě0, we have L
˜ n ÿ
k“1
akξk,
¸ ďL
˜ ÿ
kPσ
akξk,
¸ .
The following three lemmas give some useful small ball probability bounds. They correspond to [17, Lemma 3.2], [16, Corollary 2.9] and [18, Corollary 2.4] respectively.
Lemma 2.5. Letξ be a random variable with mean zero, unit variance, and finite fourth moment.
Then for every P p0,1q, there exists apP p0,1qwhich depends only onand on the fourth moment, and such that
Lpξ, q ďp.
Lemma 2.6. Let ξ1, . . . , ξn be independent centered random variables with variances at least 1 and third moments bounded byB. Then for every aPRn and everyě0, one has
L
˜ n ÿ
k“1
akξk,
¸ ď
c2 π
}a}2 `C1B ˆ}a}3
}a}2
˙3
,
where C1 is an absolute constant.
Lemma 2.7. Let A be a fixedN ˆM matrix. Consider a random vector ζ “ pζ1, . . . , ζMq where ζi are independent random variables satisfying Eζi “0, Eζi2 “1 and }ζi}ψ2 ďK. Then for every yPRN, we have
P
"
}Aζ´y}2ď1 2}A}HS
*
ď2 exp ˆ
´c}A}2HS K4}A}2
˙ .
3 Decomposition of the sphere
The rest of this paper is devoted to prove Theorem 1.2. We will make use of a partition of the unit sphere into two sets of compressible and incompressible vectors. They are first defined in [16].
Definition 3.1. Let δ ”δpNq, ρ P p0,1q. A vector xP Rn is called sparse if |supppxq| ď δn. A vector xPSn´1 is called compressible ifxis within Euclidean distance ρ from the set of all sparse vectors. A vector is called incompressible if it is not compressible. The sets of sparse, compressible and incompressible vectors in Rn will be denoted by Sparsenpδq, Compnpδ, ρq and Incompnpδ, ρq.
We sometimes neglect the subindex nwhen the dimension is clear.
Using the decompositionSn´1 “CompYIncomp, we break the invertibility problem into two subproblems, for compressible and incompressible vectors:
P
´
snpP Y ´Aq ďp? N´
?n´1q
¯ ďP
ˆ inf
xPCompnpδ,ρq
}pP Y ´Aqx}2ďp? N´
?n´1q
˙ (3.1)
`P ˆ
inf
xPIncompnpδ,ρq
}pP Y ´Aqx}2ďp? N´
?n´1q
˙ . (3.2) The bound for compressible vectors follows from the following lemma, which is a variant of Lemma 3.3 from [16].
Lemma 3.2(Invertibility for compressible vectors). There exitsN-independent constantsρ, c0, c1ą 0 such that for δďmin c1N{`
nK4logK˘ ,1(
, we have
P ˆ
inf
xPCompnpδ,ρq
}pP Y ´Aqx}2ďc0
? N
˙
ďe´c0N{K4.
Proof. We first prove a weaker result. For any fixed xPSn´1, we define the random vector ζ :“
Y x PRN. It is easy to verify that Eζi “0, Eζi2 “1 and }ζi}ψ2 ďCK. Then with }P} “ 1 and }P}2HS“N, we conclude from the Lemma 2.7 that
P
"
}pP Y ´Aqx}2ď 1 2
? N
*
ď2 exp ˆ
´cN K4
˙
. (3.3)
Let S1:“ txPSn´1:xk “0, k ąδnu. There exists an-netN inS1 of cardinality|N| ď p3{qδn. Then by (3.3) and the union bound, we get
P ˆ
DxPN :}pP Y ´Aqx}2ď1 2
? N
˙
ď2e´cN{K4` 3´1˘δn
. (3.4)
LetV be the event that}pP Y ´Aqy} ď?
N{4 for someyPS1. Recall that by the assumptions of Theorem 1.2, }P Y ´A} ďC1
?N for some constantC1. Assume thatV occurs and choose a point xPN such that}y´x} ď. Then
}pP Y ´Aqx}2ď }pP Y ´Aqy}2` }P Y ´A}}x´y}2ď1 4
?
N`C1? Nď 1
2
? N
if we choose ď1{p4C1q. Fix one such, using (3.4) we obtain that P
ˆ inf
xPS1}pP Y ´Aqx}2ď1 4
?N
˙
“PpVq ď2e´cN{K4` 3´1˘δn
ďe´c2N{K4,
if we choose c1 (and hence δ) to be sufficiently small. We use this result and take the union bound over allrδns-element subsetsσoft1, . . . , nu:
P ˆ
inf
xPSparsepδqXSn´1
}pP Y ´Aqx}2ď1 4
? N
˙
“P ˆ
Dσ,|σ| “rδns: inf
xPRσXSn´1
}pP Y ´Aqx}2ď 1 4
? N
˙
ď ˆ n
rδns
˙
e´c2N{K4 ďexp ˆ
4eδlog
´e δ
¯
n´c2N K4
˙ ďexp
ˆ
´c2N 2K4
˙
, (3.5)
with an appropriate choice ofc1, which depends only on c2.
Finally we deduce the invertibility estimate for the compressible vectors. Let c3 ą 0 and ρ P p0,1{2qto be chosen later. We need to control the eventW that }pP Y ´Aqx}2 ďc3
?N for some vectorxPComppδ, ρq. AssumeW occurs, then every such vectorxcan be written as a sumx“y`z withy PSparsepδqand}z}2ďρ. Thus}y}2ě1´ρě1{2, and
}pP Y ´Aqy}2ď }pP Y ´Aqx}2` }pP Y ´Aq}2}z}2ďc3?
N`ρC1? N .
We choosec3“1{16 andρ“1{p16C1q, so that}pP Y ´Aqy}2ď
?N{8. Since }y}2ě1{2, we can find a unit vector u“y{}y}2PSparsepδqsuch that}pP X´Aqu}2ď
?N{4. This shows that event
W implies the event in (3.5), so we havePpWq ďe´c2N{p2K4q. This concludes the proof.
Remark 3.3. If năc1N{pK4logKq, then all the vectors in Sn´1 are compressible by Definition 3.1. Hence throughout the rest of this paper, it suffices to assume
něc1N{pK4logKq. (3.6)
In particular, nis assumed to be an arbitrarily large number in the following proof.
It only remains to prove the bound for incompressible vectors in (3.2). We will divide the proof into two cases: the case where the aspect ratio λ“n{N ďλ0 for some constant 0ăλ0ă1, and the case whereλ0ăλď1. For simplicity we choose λ0“1{2 in the following, althoughλ0 can be any constant between 0 and 1. We record here an important property of the incompressible vectors, which will be used in the proof of both cases. It is Lemma 3.4 of [16].
Lemma 3.4 (Incompressible vectors are spread). Let xP Incompnpδ, ρq. Then there exists a set σ”σpxq Ď t1, . . . , nuof cardinality|σ| ě 12ρ2δn and such that
?ρ
2n ď |xk| ď 1
?δn for allkPσ. (3.7)
4 Tall matrices
In this section, we deal with the bound in (3.2) for the λă λ0 “1{2 case, i.e. P Y ´A is a tall matrix. Then it is equivalent to control the probability
P ˆ
inf
xPIncompnpδ,ρq}pP Y ´Aqx}2ďt? N
˙ .
Our proof is a modification of the one in [12].
Letxbe a vector inIncompnpδ, ρqforδ andρin Lemma 3.2. Then we take the setσgiven by Lemma 3.4. Note that every entry of Y xis of the form pY xqi“řn
k“1Yikxk, 1ďiďM, whereYik
are independent centered random variables with unit variance and bounded fourth moment. Hence we can use Lemma 2.6 and Lemma 2.4 to get that
LppY xqi, tq ď c2
π t
}Pσx}2 `C
ˆ}Pσx}3
}Pσx}2
˙3
ďC2pρq ˆ t
?δ` 1 δ? n
˙
, (4.1)
for some constant C2pρq ą0 depending only onρand the fourth moment. Here in the second step, we used the bound
}Pσx}2ě 1 2ρ2?
δ,
ˆ}Pσx}3
}Pσx}2
˙3
ď 2
ρ2δ? n,
deduced from Lemma 3.4. With (4.1) as the input, the next lemma provides a small ball probability bound forP Y x. One can refer to [19, Corollary 1.4] for the proof.
Lemma 4.1. Consider a random vector X “ pξ1, . . . , ξMq where ξi are real-valued independent random variables. Let t, pě0 such that
Lpξi, tq ďp for alli“1, . . . , M.
Let P be an orthogonal projection in RM onto an N-dimensional subspace. Then L´
P X, t? N
¯
ď pCpqN
for some absolute constantCą0.
Apply the above lemma to random vectorY x, we obtain that P
´
}pP Y ´Aqx}2ďt? N
¯ ďL´
P Y x, t? N
¯ ď
„ C3pρq
ˆ t
?δ` 1 δ? n
˙N
(4.2) for some constant C3pρq ą 0 depending only on ρ and the fourth moment. Then, as in the proof for Lemma 3.2, we can take a union bound over allxin an-net ofIncompnpδ, ρqand complete the proof by approximation.
Without loss of generality, we may assumetě1{?
δn. Otherwise, we can use the trivial bound
P
´
snpP Y ´Aq ďt? N
¯ ďP
´
snpP Y ´Aq ď pδnq´1{2? N
¯
, fortă1{? δn,
and the bound from Lemma 3.2 will dominate. Under the above assumption, thet{?
δterm in (4.2) dominates. Thus we have
P
´
}pP Y ´Aqx}2ďt? N
¯ ď
´ 2C3t{?
δ
¯N
.
By Lemma 2.2, there exists an -netN in Incompnpδ, ρqof cardinality |N| ď2np5{qn´1. Taking the union bound, we get
P
´
DxPN :}pP Y ´Aqx}2ďt? N¯
ď2n ˆ2C3t
?δ
˙Nˆ 5
˙n´1
. (4.3)
LetV be the event that}pP Y´Aqy} ďt?
N{2 for someyPIncompnpδ, ρq. Recall that}P Y ´A} ď C1
?N. Assume thatV occurs and choose a point xPN such that}x´y} ď. Then
}pP Y ´Aqx}2ď }pP Y ´Aqy}2` }P Y ´A}}x´y}2ď1 2t?
N`C1?
N ďt? N
if we choose ďt{p2C1q. Fix one such, from (4.3) we obtain that P
ˆ
inf
xPIncompnpδ,ρq}pP Y ´Aqx}2ď t 2
? N
˙
“PpVq
ď2n ˆ2C3t
?δ
˙Nˆ 10C1
t
˙n´1
ď`
C4δ´1t˘N´n`1
, (4.4)
where in the last step we usedn{Nďλ0“1{2. Together with Lemma 3.2, this concludes the proof of Theorem 1.2 for theλăλ0case.
5 Almost square matrices
In this section, we deal with the bound in (3.2) for the 1{2“λ0ăλď1 case. In particular, when λÑ1,P Y ´Abecomes analmost square matrixand (4.4) cannot provide a satisfactory probability bound. For instance, for the square matrix with N “ n, it is easy to see that the δ´N{2 term dominates over thet term. To handle this difficulty, we will use the mathods in [17], which reduce the problem of bounding}pP Y ´Aqx}2for incompressible vectorsxto a random distance problem.
We denoteN “n´1`dfor somedě1. Note that? N´?
n´1ďd{?
n. Hence to bound (3.2), it suffices to bound
P ˆ
inf
xPIncompnpδ,ρq
}pP Y ´Aqx}2ď d
?n
˙ .
We denote
m:“min
"
d, Z1
2ρ2δn
^*
. (5.1)
Let Z1 :“P Y1´A1, . . . , Zn :“ P Yn´An be the columns of the matrix Z :“ P Y ´A. Given a subsetJ Ă t1, . . . , nuof cardinalitym, we define the subspaces
HJc :“spanpZkqkPJc ĂRN. (5.2)
For levelsK1“ρa
δ{2 andK2“K1´1, we define the set of totally spread vectors SJ:“
"
yPSpRJq: K1
?m ď |yk| ď K2
?m for allkPJ
*
. (5.3)
In the following lemma, we letJ be a random subset uniformly chosen over all subsets oft1, . . . , nu of cardinality m. We shall writePJ forPRJ, the orthogonal projection onto the subspace RJ, and denote the probability and expectation over the random subset J byPJ andEJ.
Lemma 5.1. There exists constantc2ą0depending only onρsuch that for everyxPIncompnpδ, ρq, the event
Epxq:“
"
PJx
}PJx}2 PSJ and ρ?
? m
2n ď }PJx}2ď
?m
?δn
*
satisfies PJpEpxqq ě pc2δqm.
Proof. LetσĂ t1, . . . , nube the subset from Lemma 3.4. Then we have PJpJ Ăσq “
ˆ|σ|
m
˙ {
ˆn m
˙ .
Using Stirling’s approximation, fordď 14ρ2δn, we have PJpJ Ăσq ě
ˆc|σ|
n
˙m
ě pc2δqm,
and fordą 14ρ2δn, we have
PJpJĂσq ě ˆn
m
˙´1
ě m!
nm ě
´cm n
¯m
ě pc2δqm.
IfJ Ăσ, then summing (3.7) overkPJ, we obtain the required two-sided bound for }PJx}2. This and (3.7) yieldPJx{}PJx}2PSJ. Hence Epxqholds.
Lemma 5.1 implies the following lemma, whose proof is a minor modification of the one for [17, Lemma 6.2].
Lemma 5.2. LetJ denote the m-element subsets of t1, . . . , nu. Then for every ą0, P
ˆ
inf
xPIncompnpδ,ρq}Zx}2ăρ cm
2n
˙
ď pc2δq´mmax
J P ˆ
inf
xPSJdistpZx, HJcq ă
˙
. (5.4) It remains to boundPpinfxPSJdistpZx, HJcq ăqfor anym-element subsetJ.
Theorem 5.3 (Uniform distance bound). Let Y be a random matrix satisfying the assumptions in Theorem 1.2. Then for any m-element subset J and anytą0, there existC, cą0 independent of J such that
P ˆ
inf
xPSJ
distpZx, HJcq ăt? d
˙
ď pCtK5logKqd`e´cN. (5.5) By the definition ofmin (5.1), we have
pc2δq´mď
”
pc2δq´ρ2δ{2 ın
ďe´cN{2, (5.6)
where we used thatδ“c1{pK4logKqfor some sufficiently smallc1ą0 (see Lemma 3.2). Then we conclude from Theorem 5.3 and Lemma 5.2 that
P
˜
inf
xPIncompnpδ,ρq
}pP Y ´Aqx}2ăρ
?md
?n
¸
ďpc2δq´mpCK5logKqd` pc2δq´me´cN ď`
C1K9plogKq2˘d
`e´cN{2, (5.7) where in the last step we used (5.6),mďdandd{mďCK4logK. Hence we obtain from (5.7) that
P ˆ
inf
xPIncompnpδ,ρq
}pP Y ´Aqx}2ăρ d
?n
˙ ď
´
C1K11plogKq5{2
¯d
`e´cN{2,
which, together with Lemma 3.2, concludes the proof of Theorem 1.2.
To prove Theorem 5.3, we need the following lemma, which gives a lower bound for the distance between a single random vector and a random subspace. We will prove it in Section 6. Note that by choosing small enoughρandδin (5.1), we can assume m`d´1ď3N{4.
Lemma 5.4 (Distance to a random subspace). Let J be any m-element subset of t1, . . . , nu and let HJc be the random subspace of RN as defined in (5.2). Let X be a random vector in RM whose coordinates are i.i.d. centered random variables with unit variance and finite fourth moment, independent of HJc. Let l“m`d´1such that lď3N{4. Then for everyą0, we have
P ˆ
sup
vPRN
P
´
distpP X´v, HJcq ă? l
ˇ ˇ ˇHJc
¯
ą pCql`e´cN
˙
ďe´cN, (5.8) where C, cą0 depend only on the fourth moment of the entries ofX. Moreover, (5.8) implies the following weaker result:
sup
vPRN
P
´
distpP X´v, HJcq ă? l
¯
ď pCql`e´cN. (5.9) Note that for any fixed x P SJ, we have Zx “ P Y x´Ax, where Y x is a random vector that satisfies the assumption for X in Lemma 5.4. So (5.9) gives a useful probability bound for a single x P SJ. We might then take a union bound over all z in an -net of SJ and complete by approximation. Before proving Theorem 5.3, we make several reductions. First, we can assume that the entries of Y have absolute continuous distributions. In fact we can add to each entry an independent Gaussian random variable with small varianceσ, and later letσÑ0 (all the estimates below do not depend on σ). The main purpose of this reduction is to have the following equality
dimpHJcq “n´m a.s. (5.10)
Second, without loss of generality, in the proof of (5.5) we can assume
těe´c1N{d, for some smallc1ą0. (5.11) LetPHK be the orthogonal projection inRN ontoHJKc, and define
W :“ PHKP Y|RJ. (5.12)
Then for every xPRn, we have
distpP Y x´v, HJcq “ }W x´w}2, wherew“PHKv. (5.13) By (5.10), dimpHJKcq “ N ´n`m “ l almost surely. Thus W acts as an operator from an m- dimensional subspace into an l-dimensional subspace almost surely. If we have a proper operator bound for W, we can run the approximation argument on the -net of SJ and prove a uniform distance bound over allxPSJ.
Lemma 5.5. Let W be a random matrix as in (5.12) and let w be a random vector as in (5.13).
For every t satisfying (5.11) and everyC0ą0, we have P
ˆ inf
xPSJ}W x´w}2ăt?
d,}W} ďC0K? d
˙
ďKm´1pC1tqd.
Proof. Fix anyxPSJ. It is easy to verify thatY xis a random vector that satisfies the assumption forX in Lemma 5.4, which implies
P
´
}W x´w}2ăt? d
¯ ďP
´
distpP X´v, HJcq ăt? l
¯
ď pCtqm`d´1, (5.14)
fort satisfying (5.11).
Let“t{pC0Kq. By Lemma 2.2, there exists an-netN of SJĂSpRJqof cardinality
|N| ď2m ˆ
1`2
˙m´1
ď2m
ˆ3C0K t
˙m´1
.
Consider the event
E:“
"
xPNinf }W x´w}2ă2t? d
* .
Taking union bound, we get
PpEq ď |N|Pp}W x´w}2ď2t?
dq ď2m
ˆ3C0K t
˙m´1
pCtqm`d´1ďKm´1pC1tqd´1. (5.15) Suppose there exists yPSJ such that
}W y´w}2ăt?
d, }W} ďC0K? d.
Then we choose xPN such that}x´y}2ď, and by triangle inequality we obtain that }W x´w}2ď }W y´w}2` }W}}x´y}2ăt?
d`C0K?
dď2t? d,
i.e. the eventE holds. Then the bound (5.15) concludes the proof.
To bound the norm ofW, we have the following proposition.
Proposition 5.6. LetW be a random matrix as in (5.12). Then P
´
}W} ěsK? d
ˇ ˇ ˇHJc
¯
ďe´˜cs2d, forsěC,˜ where C,˜ ˜cą0 are absolute constants.
Proof. For simplicity of notations, we fix an HJc during the proof and omit the conditioning on it from the expressions below. LetN be anp1{2q-net ofSpRJqandMbe anp1{2q-net ofSpHJKcq. By Lemma 2.2, we can choose N andMsuch that
|N| ď2m¨5m´1, |M| ď2l¨5l´1. It is easy to prove that
}W} ď4 sup
xPN,yPM
|xW x, yy|. (5.16)
For everyxPN andyPM,xW x, yy “ xP Y x, yy “ xY x, PTyyis a random variable with subgaussian moment bounded by CK for some absolute constantCą0. Hence we have
P
´
|xW x, yy| ěsK? d
¯
ď2e´cs2d. Using (5.16) and taking the union bound, we get
P
´
}W} ěsK? d
¯
ď32ml¨5m´1¨5l´1¨e´cs2dďe´˜cs2d,
where in the last step we used mďlď2dandsěC˜ for some sufficiently large ˜Cą0.
With Lemma 5.5 and Proposition 5.6, we have P
ˆ inf
xPSJ
}W x´w}2ăt? d
˙
ďKm´1pC1tqd`e´˜cC02d.
Unfortunately, the bounde´˜cC02dis too weak for smalld. Following the idea in [17], we refine Lemma 5.5 by decoupling the information about}W x´w}2 from the information about}W}.
Lemma 5.7 (Decoupling). Let W be an N ˆm matrix whose columns are independent random vectors, and let w be an arbitrary fixed vector in RN. Let zPSJ, which satisfies zk ěK1{?
m for all 1ďkďm (see (5.3)). Then for every0ăaăb, we have
Pp}W z´w}2ăa,}W} ąbq ď2 sup
xPSm´1,vPRN
P ˆ
}W x´v}2ă
?2a K1
˙ P
ˆ
}W} ą b
?2
˙ .
Proof. Ifm“1, then }W} “ }W z}2, so the probability on the left hand side is zero. Hence in the following we only considermě2. Then we can decompose the index sett1, . . . , muinto two disjoint subsetsIandH with|I| “rm{2s. We writeW “WI`WH, whereWI andWH are the submatrices of W with columns inI and H, respectively. Similarly, we write z “ zI `zH for z PSJ. Using }W}2ď }WI}2` }WH}2, we have
Pp}W z´w}2ăa,}W} ąbq ďpI`pH,
where
pI “P
´
}W z´w}2ăa,}WH} ąb{? 2¯
“P
´
}W z´w}2ăa ˇ ˇ
ˇ}WH} ąb{? 2
¯ P
´
}WH} ąb{? 2
¯ ,
and similarly for pH. We shall boundpI in the following, while the arguments forpH are similar.
WritingW z“WIzI `WHzH and using the independence ofWI andWH, we conclude that pI ď sup
uPRN
Pp}WIzI´u}2ăaqP ˆ
}WH} ą b
?2
˙
ď sup
uPRN
Pp}W zI´u}2ăaqP ˆ
}W} ą b
?2
˙
. (5.17)
By the assumption on z and using|I| ěm{2, we have }zI}2ěK1{
?2.Hence for x“zI{}zI}2and v“u{}zI}2, we obtain
Pp}W zI´u}2ăaq ďP
´
}W x´v}2ă
?2a{K1¯ .
Together with (5.17), this concludes the proof.
With this decoupling lemma, we can prove the following refinement of Lemma 5.5.
Lemma 5.8. Let W be a random matrix as in (5.12) and let w be a random vector as in (5.13).
For every są1, every C0ą0 and everyt satisfying (5.11), we have P
ˆ inf
xPSJ
}W x´w}2ăt?
d andsC0K?
dă }W} ď2sC0K? d
˙
ďKm´1K1´pm`d´1q´
C2te´c2s2¯d
.
Proof. Let“t{p2sC0Kq. By Lemma 2.2, there exists an-netN ofSJ ĂSpRJqof cardinality
|N| ď2m ˆ
1`4
˙m´1
ď2m
ˆ9sC0K t
˙m´1
.
Consider the event
E:“
"
xPNinf }W x´w}2ă2t?
dand}W} ąsC0K? d
* .
Note that conditioning onHJc,W andw satisfy the assumptions in Lemma 5.7. Using Lemma 5.7 and taking union bound, we get
PpE|HJcq ď |N| ¨2 sup
xPSd´1,vPRN
P ˆ
}W x´v}2ă
?2 K1
¨2t? d
ˇ ˇ ˇ ˇ
HJc
˙ P
˜
}W} ě sC0K?
? d 2
ˇ ˇ ˇ ˇ ˇ
HJc
¸
Taking expectation over HJc and using Proposition 5.6, we obtain that PpEq ď4m
ˆ9sC0K t
˙m´1
e´c1s2dE
« sup
xPSd´1,vPRN
P ˆ
}W x´v}2ă
?2 K1
¨2t? d
ˇ ˇ ˇ ˇ
HJc
˙ˇ ˇ ˇ ˇ ˇ
HJc
ff
ď4m
ˆ9sC0K t
˙m´1
e´c1s2d ˆC1t
K1
˙m`d´1
ďKm´1K1´pm`d´1q´
C2te´c2s2¯d
, (5.18) where in the second step we used the representation in (5.13), (5.8) and (5.11), and in the last step we used sě1 and 1ďmďd.
Suppose there existsyPSJ such that }W y´w}2ăt?
d and sC0K?
dă }W} ď2sC0K? d.
Then we choose xPN such that}x´y}2ď, and by triangle inequality we obtain that }W x´w}2ď }W y´w}2` }W}}x´y}2ăt?
d`2sC0K?
dď2t? d,
i.e. the eventE holds. The bound (5.18) concludes the proof.
Proof of Theorem 5.3. Recall that we assume (5.11) holds. Summing the probability bounds in Lemma 5.5 and Lemma 5.8 fors“2k,kPZ`, we conclude that
P ˆ
inf
xPSJ}W x´w}2ăt? d
˙
ďKm´1pC1tqd` ÿ
s“2k,kPZ`
Km´1K1´pm`d´1q
´
C2te´c2s2
¯d
ď`
C3KK1´2t˘d .
Recall that K1“ρa
δ{2 (see (5.3)) andδ“c1N{pnK4logKq(see Lemma 3.2). Then we get that P
ˆ inf
xPSJ
}W x´w}2ăt? d
˙ ď`
C3tK5logK˘d
.
This concludes the proof.
6 Proof of Lemma 5.4
We will first prove a general inequality that holds for any fixed subspaceH. This probability bound depends on the arithmetic structure of the subspace, which can be expressed using theleast common denominator (LCD). Then we will prove a bound for the LCD of the random subspaceHJc.
Forαą0 andγP p0,1q, the least common denominator of a vectoraPRM is defined as LCDα,γpaq:“inf θą0 : distpθa,ZMq ăminpγ}θa}2, αq(
.
The above definition can be generalized to higher dimensions. Let a“ pa1, . . . , aMqbe a sequence of vectorsak PRl. We define the product of the multi-vectoraand a vectorθPRl as
θ¨a“ pxθ, a1y, . . . ,xθ, aMyq PRM.
Then we define, for αą0 andγP p0,1q,
LCDα,γpaq:“inf }θ}2:θPRl,distpθ¨a,ZMq ăminpγ}θ¨a}2, αq( .
Finally forαą0 andγP p0,1q, the least common denominator of asubspaceEĂRM is defined as LCDα,γpEq:“inftLCDα,γpaq:aPSpEqu “inf }θ}2:θPE,distpθ,ZMq ăminpγ}θ}2, αq(
. (6.1) A key to the proof is the next small ball probability theorem. It is stated as Theorem 3.3 in [17].
Theorem 6.1. Consider a sequencea“ pa1, . . . , aMqof vectorsak PRl, which satisfies
M
ÿ
k“1
xx, aky2ě }x}22 for every xPRl. (6.2) Let ξ1, . . . , ξM be independent and identically distributed, mean zero random variables, such that Lpξk,1q ď1´b for some b ą0. Consider the random sum S “řM
k“1akξk PRl. Then for every αą0 andγP p0,1q, and for
ě
?l
LCDα,γpaq, we have
LpS, ? lq ď
ˆ C γ?
b
˙l
`Cle´2bα2.
LetH be a fixed subspace inRN of dimensionN´l. We denote an orthonormal basis ofHKby tn1, . . . , nlu ĂRN, and write X in coordinates asX “ pξ1, . . . , ξMq. Then usingP PT “1, we get
distpP X´v, Hq “ }PHKpP X´vq}2“
›
›
›
›
›
r
ÿ
r“1
xP X, nrynr´PHKv
›
›
›
›
›2
“
›
›
›
›
›
r
ÿ
r“1
xX, PTnrynr´PHKv
›
›
›
›
›2
“
›
›
›
›
›
r
ÿ
r“1
xX, PTnryPTnr´PTPHKv
›
›
›
›
›2
“ }PEX´w}2“
›
›
›
›
›
M
ÿ
k“1
akξk´w
›
›
›
›
›2
,
where E”EpHq:“PTHK,ak“PEek,k“1, . . . , M, andw“PTPHKvPRM. Notice that
M
ÿ
k“1
xx, aky2“ }x}22, for anyxPE.
Hence the subsequence of vectors a “ pa1, . . . , aMq satisfies the assumption (6.2). So we can use Theorem 6.1 in the space E, which is identified with Rl by a suitable isometry. For every θ “ pθ1, . . . , θMq PE and everyk, we havexθ, aky “ xθ, eky “θk. Soθ¨a“θ, where the right hand side is considered as a vector in RM. Therefore, we have
LCDα,γpEq “LCDα,γpaq.
By Lemma 2.5, we have Lpξk,1{2q ď1´bfor some bą0 that depends only on the fourth moment ofξk. Hence we can apply Theorem 6.1 to S“řM
k“1akξk and conclude that for allą0, P
´
distpP X´v, Hq ă? l
¯
ďLpS, ? lq ď
ˆC γ
˙l
`
˜
C? l γLCDα,γpEq
¸l
`Cle´cα2. (6.3) Applying the above inequality to the random subspacesHJc andE“EpHJcq, we conclude Lemma 5.4 if we can prove the following theorem. Heuristically, it shows that the randomness will remove any arithmetic structure from the subspace Eand make the LCD exponentially large.
Theorem 6.2. Supposeξ1, . . . , ξN´l are independent centered random variables with unit variance and uniformly bounded fourth moment. Let Y˜ be an M ˆ pN ´lq random matrix whose rows are independent copies of the random vectorpξ1, . . . , ξN´lq, andA˜be a fixedNˆ pN´lqmatrix. Assume that }Y˜} ` }A} ď˜ C1
?N for some constantC1ą0. LetH be the random subspace of RN spanned by the column vectors of PY˜ ´A, and˜ E”EpHqbe the random subspacePTHK ofRM. Then for α“c?
N, we have
P
´
LCDα,cpEq ăc?
N ecN{l¯
ďe´cN,
where cdepends only on the maximal fourth moment.
The rest of this section is devoted to proving this theorem. Note that ifaPEpHq, thena“PTx for some xPHK. On the other hand, usingx“P awe obtain that
xPHKôY˜TPTx´A˜Tx“0ôY˜Ta´A˜TP a“0.
We define thepN´lq ˆM matrix ˜B :“A˜TP. Then for every setS in E, we have
xPSinf
›
›
›Y˜Tx´Bx˜
›
›
›2ą0 impliesSXE“ H. (6.4)
This helps us to navigate the random subspaceE away from undesired setsS on the unit sphere.
As in Definition 3.1, we can define the compressible and incompressible vectors onSM´1, which are denoted byCompMpδ, ρqandIncompMpδ, ρq, respectively. We can prove the following result for compressible vectors as in Lemma 3.2.
Lemma 6.3(Random subspaces are incompressible). There existN-independent constantsρ, c0, c1ą 0 such that for 0ăδďc1, we have
P ˆ
inf
xPCompMpδ,ρq
›
›
›
´Y˜T ´B˜
¯ x
›
›
›2ďc0
? N
˙
ďe´c0N. (6.5)
In particular, this implies that P`
EXSM´1ĎIncompMpδ, ρq˘
ě1´e´c0N. (6.6)
Proof. The proof is similar to the one for Lemma 3.2. The only difference is that instead of using Lemma 2.7, we shall use that ˜YT has independent row vector ˜Y1, . . . ,Y˜N´l. For anyxPSM´1, it is easy to verify thatxY˜k, xyhave variance 1 and uniformly bounded fourth moment. Then by Lemma 2.5, we have for any fixedv“ pv1, . . . , vN´lq PRN´l,
P
´
|xY˜k, xy ´vk| ď1{2
¯ ďp
for some 0ăpă1. Combining with the tensorization Lemma 2.1, we can find constantsη, νP p0,1q depending onpsuch that
P
!
}Y˜Tx´v}2ďη? N´l
)
ďνN´l. (6.7)
Note thatl ď3N{4 andM ďCN by our assumptions. Then using estimate (6.7) instead of (3.3), we can complete the rest of the proof as in Lemma 3.2.
Fixδ and ρ given by Lemma 6.3 for the rest of this section. Note that we can take δ to be a constant of order 1 (in contrast toδďc1N{`
nK4logK˘
in Lemma 3.2). We will further decompose IncompMpδ, ρq into level setsSD according to the value D of the least common denominator. We shall prove a nontrivial lower bound on infxPSD}pY˜T ´Bqx}˜ 2 for each level set up to D of the exponential order. By (6.4), this means that E is disjoint from every such level set. Therefore, E must have exponentially large least common denominators. First, as a consequences of Lemma 3.4, we have the following lemma, which gives a weak lower bound for the LCD. It is Lemma 3.6 of [17].
Lemma 6.4. For every δ, ρP p0,1q, there exists c2pδ, ρq ą0 and c3pδq ą0 such that the following holds. Let aPIncompMpδ, ρq. Then for every 0ăcăc2pδ, ρqand everyαą0, one has
LCDα,cpaq ąc3pδq? M .
We then decomposeIncompMpδ, ρqaccording to the values of LCD.
Definition 6.5 (Level sets). Let Děc3pδq?
M. DefineSDĎSM´1 as SD:“ txPIncompMpδ, ρq:DďLCDα,cpxq ă2Du X`
PTRN˘ .
To obtain a lower bound for}pY˜T ´Bqx}˜ 2 onSD, we first prove a bound for a single vectorx and then use an-net argument.
Lemma 6.6. LetxPSD. Then for every tą0 we have P
´
}pY˜T ´B˜qx}2ăt? N
¯ ď
ˆ Ct`C
D `Ce´cα2
˙N´l
.
Proof. Denoting the elements of ˜YT byξjk, we can write thej-th coordinate of ˜YTxas
´Y˜Tx
¯
j“
M
ÿ
k“1
ξjkxk“:ζj, j“1, . . . , N´l.
We denote the coordinates of ˜Bxbyvj, 1ďj ďN´l. Applying Theorem 6.1 in dimension 1, we obtain that for every j and everytą0,
Pp|ζj´vj| ătq ďCt` C
LCDα,cpxq`Ce´cα2ďCt`C
D `Ce´cα2.
