arXiv subjects

Alex Samorodnitsky

Publications and source records attributed to Alex Samorodnitsky.

At least 19 recordsLinked to original sources

arXiv2024

Eigenvalues and eigenfunctions of a Hamming ball

We describe the eigenvalues and the eigenspaces of the adjacency matrices of subgraphs of the Hamming cube induced by Hamming balls, and more generally, by a union of adjacent concentric Hamming spheres. As a corollary, we extend the range of cardinalities of subsets of the Hamming cube for which Hamming balls have essentially the largest maximal eigenvalue (among all subsets of the same size). We show that this holds even when the sets in question are large, with cardinality which is an arbitrary subconstant fraction of the whole cube.

math.CO↗

arXiv2023

Optimal Discrimination Between Two Pure States and Dolinar-Type Coherent-State Detection

We consider the problem of discrimination between two pure quantum states. It is well known that the optimal measurement under both the error-probability and log-loss criteria is a projection, while under an ``erasure-distortion'' criterion it is a three-outcome positive operator-valued measure (POVM). These results were derived separately. We present a unified approach which finds the optimal measurement under any distortion measure that satisfies a convexity relation with respect to the Bhattacharyya distance. Namely, whenever the measure is relatively convex (resp. concave), the measurement is the projection (resp. three-outcome POVM) above. The three above-mentioned results are obtained as special cases of this simple derivation. As for further measures for which our result applies, we prove that Renyi entropies of order $1$ and above (resp. $1/2$ and below) are relatively convex (resp. concave). A special setting of great practical interest, is the discrimination between two coherent-light waveforms. In a remarkable work by Dolinar it was shown that a simple detector consisting of a photon counter and a feedback-controlled local oscillator obtains the quantum-optimal error probability. Later it was shown that the same detector (with the same local signal) is also optimal in the log-loss sense. By applying a similar convexity approach, we obtain in a unified manner the optimal signal for a variety of criteria.

quant-ph↗

arXiv2023

On the difficulty to beat the first linear programming bound for binary codes

The first linear programming bound of McEliece, Rodemich, Rumsey, and Welch is the best known asymptotic upper bound for binary codes, for a certain subrange of distances. Starting from the work of Friedman and Tillich, there are, by now, some arguably easier and more direct arguments for this bound. We show that this more recent line of argument runs into certain difficulties if one tries to go beyond this bound (say, towards the second linear programming bound of McEliece, Rodemich, Rumsey, and Welch).

cs.IT↗

arXiv2022

On some properties of random and pseudorandom codes

We describe some pseudorandom properties of binary linear codes achieving capacity on the binary erasure channel under bit-MAP decoding (as shown in Kudekar et al this includes doubly transitive codes and, in particular, Reed-Muller codes). We show that for all integer $q \ge 2$ the $\ell_q$ norm of the characteristic function of such 'pseudorandom' code decreases as fast as that of any code of the same rate (and equally fast as that of a random code) under the action of the noise operator. In information-theoretic terms this means that the $q^{th}$ Rényi entropy of this code increases as fast as possible over the binary symmetric channel. In particular (taking $q = \infty$) this shows that such codes have the smallest asymptotic undetected error probability (equal to that of a random code) over the BSC, for a certain range of parameters. We also study the number of times a certain local pattern, a 'rhombic' $4$-tuple of codewords, appears in a linear code, and show that for a certain range of parameters this number for pseudorandom codes is similar to that for a random code.

cs.IT↗

arXiv2022

Weight distribution of random linear codes and Krawchouk polynomials

For $0 < λ< 1$ and $n \rightarrow \infty$ pick uniformly at random $λn$ vectors in $\{0,1\}^n$ and let $C$ be the orthogonal complement of their span. Given $0 < γ< \frac12$ with $0 < λ< h(γ)$, let $X$ be the random variable that counts the number of words in $C$ of Hamming weight $i = γn$ (where $i$ is assumed to be an even integer). Linial and Mosheiff determined the asymptotics of the moments of $X$ of all orders $o\left(\frac{n}{\log n}\right)$. In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized $k^{th}$ moment of $X$ is essentially determined by the $k^{th}$ norm of the Krawchouk polynomial $K_i$.

math.CO↗

arXiv2022

On the Round Complexity of Randomized Byzantine Agreement

We prove lower bounds on the round complexity of randomized Byzantine agreement (BA) protocols, bounding the halting probability of such protocols after one and two rounds. In particular, we prove that: (1) BA protocols resilient against $n/3$ [resp., $n/4$] corruptions terminate (under attack) at the end of the first round with probability at most $o(1)$ [resp., $1/2+ o(1)$]. (2) BA protocols resilient against a fraction of corruptions greater than $1/4$ terminate at the end of the second round with probability at most $1-Θ(1)$. (3) For a large class of protocols (including all BA protocols used in practice) and under a plausible combinatorial conjecture, BA protocols resilient against a fraction of corruptions greater than $1/3$ [resp., $1/4$] terminate at the end of the second round with probability at most $o(1)$ [resp., $1/2 + o(1)$]. The above bounds hold even when the parties use a trusted setup phase, e.g., a public-key infrastructure (PKI). The third bound essentially matches the recent protocol of Micali (ITCS'17) that tolerates up to $n/3$ corruptions and terminates at the end of the third round with constant probability.

cs.CR↗

arXiv2021

Hypercontractive inequalities for the second norm of highly concentrated functions, and Mrs. Gerber's-type inequalities for the second Renyi entropy

Let $T_ε$, $0 \le ε\le 1/2$, be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a distribution on $\{0,1\}^n$ and let $q > 1$. We prove tight Mrs. Gerber-type results for the second Renyi entropy of $T_ε f$ which take into account the value of the $q^{th}$ Renyi entropy of $f$. For a general function $f$ on $\{0,1\}^n$ we prove tight hypercontractive inequalities for the $\ell_2$ norm of $T_ε f$ which take into account the ratio between $\ell_q$ and $\ell_1$ norms of $f$.

cs.IT↗

arXiv2021

One more proof of the first linear programming bound for binary codes and two conjectures

We give one more proof of the first linear programming bound for binary codes, following the line of work initiated by Friedman and Tillich. The new argument is somewhat similar to previous proofs, but we believe it to be both simpler and more intuitive. Moreover, it provides the following 'geometric' explanation for the bound. A binary code with minimal distance $δn$ is small because the projections of the characteristic functions of its elements on the subspace spanned by the Walsh-Fourier characters of weight up to $\left(\frac 12 - \sqrt{δ(1-δ)}\right) \cdot n$ are essentially independent. Hence the cardinality of the code is bounded by the dimension of the subspace. We present two conjectures, suggested by the new proof, one for linear and one for general binary codes which, if true, would lead to an improvement of the first linear programming bound. The conjecture for linear codes is related to and is influenced by conjectures of Håstad and of Kalai and Linial. We verify the conjectures for the (simple) cases of random linear codes and general random codes.

cs.IT↗

arXiv2020

An improved bound on $\ell_q$ norms of noisy functions

Let $T_ε$, $0 \le ε\le 1/2$, be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$ and let $q \ge 1$. In arXiv:1809.09696 the $\ell_q$ norm of $T_ε f$ was upperbounded by the average $\ell_q$ norm of conditional expectations of $f$, given sets whose elements are chosen at random with probability $λ$, depending on $q$ and on $ε$. In this note we prove this inequality for integer $q \ge 2$ with a better (smaller) parameter $λ$. The new inequality is tight for characteristic functions of subcubes. As an application, following arXiv:2008.07236, we show that a Reed-Muller code $C$ of rate $R$ decodes errors on $\mathrm{BSC}(p)$ with high probability if \[ R ~<~ 1 - \log_2\left(1 + \sqrt{4p(1-p)}\right). \] This is a (minor) improvement on the estimate in arXiv:2008.07236.

cs.IT↗

arXiv2020

On codes decoding a constant fraction of errors on the BSC

Using techniques and results from Kudekar et al. we strengthen the bounds on the weight distribution of linear codes achieving capacity on the BEC, which were shown by the first author. In particular, we show that for any doubly transitive binary linear code $C \subseteq \{0,1\}^n$ of rate $0 < R < 1$ with weight distribution $\left(a_0,...,a_n\right)$ holds $a_i \le 2^{o(n)} \cdot (1-R)^{-2 \ln 2 \cdot \min\{i, n-i\}}$. For doubly transitive codes with minimal distance at least $Ω\left(n^c\right)$, $0 < c \le 1$, the error factor of $2^{o(n)}$ in this bound can be removed at the cost of replacing $1-R$ with a smaller constant $a = a(R,c) < 1- R$. Moreover, in the special case of Reed-Muller codes, due to the additional symmetries of these codes, this error factor can be removed at essentially no cost. This implies that for any doubly transitive code $C$ of rate $R$ with minimal distance at least $Ω\left(n^c\right)$, there exists a positive constant $p = p(R,c)$ such that $C$ decodes errors on $\mathrm{BSC}(p)$ with high probability if $p < p(R,c)$. For doubly transitive codes of a sufficiently low rate (smaller than some absolute constant) the requirement on the minimal distance can be omitted, and hence this critical probability $p(R)$ depends only on $R$. Furthermore, $p(R) \rightarrow \frac12$ as $R \rightarrow 0$. In particular, a Reed-Muller code $C$ of rate $R$ decodes errors on $\mathrm{BSC}(p)$ with high probability if \[ R ~<~ 1 - \big(4p(1-p)\big)^{\frac{1}{4 \ln 2}}, \] answering a question of Abbe, Hazla, and Nachum.

cs.IT↗

arXiv2019

An upper bound on $\ell_q$ norms of noisy functions

Let $T_ε$ be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$ and let $q \ge 1$. We upper bound the $\ell_q$ norm of $T_ε f$ by the average $\ell_q$ norm of conditional expectations of $f$, given sets of roughly $(1-2ε)^{r(q)} \cdot n$ variables, where $r$ is an explicitly defined function of $q$. We describe some applications for error-correcting codes and for matroids. In particular, we derive an upper bound on the weight distribution of duals of BEC-capacity achieving binary linear codes. This improves the known bounds on the linear-weight components of the weight distribution of constant rate binary Reed-Muller codes for almost all rates.

cs.IT↗

arXiv2019

A moment ratio bound for polynomials and some extremal properties of Krawchouk polynomials and Hamming spheres

Let $p \ge 2$. We improve the bound $\frac{\|f\|_p}{\|f\|_2} \le (p-1)^{s/2}$ for a polynomial $f$ of degree $s$ on the boolean cube $\{0,1\}^n$, which comes from hypercontractivity, replacing the right hand side of this inequality by an explicit bivariate function of $p$ and $s$, which is smaller than $(p-1)^{s/2}$ for any $p > 2$ and $s > 0$. We show the new bound to be tight, within a smaller order factor, for the Krawchouk polynomial of degree $s$. This implies several nearly-extremal properties of Krawchouk polynomials and Hamming spheres (equivalently, Hamming balls). In particular, Krawchouk polynomials have (almost) the heaviest tails among all polynomials of the same degree and $\ell_2$ norm (this has to be interpreted with some care). The Hamming spheres have the following approximate edge-isoperimetric property: For all $1 \le s \le \frac{n}{2}$, and for all even distances $0 \le i \le \frac{2s(n-s)}{n}$, the Hamming sphere of radius $s$ contains, up to a multiplicative factor of $O(i)$, as many pairs of points at distance $i$ as possible, among sets of the same size (there is a similar, but slightly weaker and somewhat more complicated claim for general distances). This also implies that Hamming spheres are (almost) stablest with respect to noise among sets of the same size. In coding theory terms this means that a Hamming sphere (equivalently a Hamming ball) has the maximal probability of undetected error, among all binary codes of the same rate. We also describe a family of hypercontractive inequalities for functions on $\{0,1\}^n$, which improve on the `usual' "$q \rightarrow 2$" inequality by taking into account the concentration of a function (expressed as the ratio between its $\ell_r$ norms), and which are nearly tight for characteristic functions of Hamming spheres.

math.CO↗

arXiv2019

Improved log-Sobolev inequalities, hypercontractivity and uncertainty principle on the hypercube

Log-Sobolev inequalities (LSIs) upper-bound entropy via a multiple of the Dirichlet form (i.e. norm of a gradient). In this paper we prove a family of entropy-energy inequalities for the binary hypercube which provide a non-linear comparison between the entropy and the Dirichlet form and improve on the usual LSIs for functions with small support. These non-linear LSIs, in turn, imply a new version of the hypercontractivity for such functions. As another consequence, we derive a sharp form of the uncertainty principle for the hypercube: a function whose energy is concentrated on a set of small size, and whose Fourier energy is concentrated on a small Hamming ball must be zero. The tradeoff between the sizes that we derive is asymptotically optimal. This new uncertainty principle implies a new estimate on the size of Fourier coefficients of sparse Boolean functions. We observe that an analogous (asymptotically optimal) uncertainty principle in the Euclidean space follows from the sharp form of Young's inequality due to Beckner. This hints that non-linear LSIs augment Young's inequality (which itself is sharp for finite groups).

math.PR↗

arXiv2018

On $\ell_4 : \ell_2$ ratio of functions with restricted Fourier support

Given a subset $A \subseteq \{0,1\}^n$, let $μ(A)$ be the maximal ratio between $\ell_4$ and $\ell_2$ norms of a function whose Fourier support is a subset of $A$. We make some simple observations about the connections between $μ(A)$ and the additive properties of $A$ on one hand, and between $μ(A)$ and the uncertainty principle for $A$ on the other hand. One application obtained by combining these observations with results in additive number theory is a stability result for the uncertainty principle on the discrete cube. Our more technical contribution is determining $μ(A)$ rather precisely, when $A$ is a Hamming sphere $S(n,k)$ for all $0 \le k \le n$.

math.CO↗

arXiv2018

On coset leader graphs of structured linear codes

We suggest a new approach to obtain bounds on locally correctable and some locally testable binary linear codes, by arguing that these codes (or their subcodes) have coset leader graphs with high discrete Ricci curvature. The bounds we obtain for locally correctable codes are worse than the best known bounds obtained using quantum information theory, but are better than those obtained using other methods, such as the "usual" information theory. (We remark that our methods are completely elementary.) The bounds we obtain for a family of locally testable codes improve the best known bounds.

math.CO↗

arXiv2016

Hafnians, perfect matchings and Gaussian matrices

We analyze the behavior of the Barvinok estimator of the hafnian of even dimension, symmetric matrices with nonnegative entries. We introduce a condition under which the Barvinok estimator achieves subexponential errors, and show that this condition is almost optimal. Using that hafnians count the number of perfect matchings in graphs, we conclude that Barvinok's estimator gives a polynomial-time algorithm for the approximate (up to subexponential errors) evaluation of the number of perfect matchings.

math.PR↗

arXiv2016

On the entropy of a noisy function

Let $0 < ε< 1/2$ be a noise parameter, and let $T_ε$ be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$. We upper bound the entropy of $T_ε f$ by the average entropy of conditional expectations of $f$, given sets of roughly $(1-2ε)^2 \cdot n$ variables. In information-theoretic terms, we prove the following strengthening of "Mrs. Gerber's lemma": Let $X$ be a random binary vector of length $n$, and let $Z$ be a noise vector, corresponding to a binary symmetric channel with crossover probability $ε$. Then, setting $v = (1-2ε)^2 \cdot n$, we have (up to lower-order terms): $$ H\Big(X \oplus Z\Big) \ge n \cdot H\left(ε~+~ (1-2ε) \cdot H^{-1}\left(\frac{{\mathbb E}_{|B| = v} H\Big(\{X_i\}_{i\in B}\Big)}{v}\right)\right) $$ As an application, we show that for a boolean function $f$, which is close to a characteristic function $g$ of a subcube of dimension $n-1$, the entropy of $T_ε f$ is at most that of $T_ε g$. This, combined with a recent result of Ordentlich, Shayevitz, and Weinstein shows that the "Most informative boolean function" conjecture of Courtade and Kumar holds for high noise $ε\ge 1/2 - δ$, for some absolute constant $δ> 0$. Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $ε$, then, provided $ε\ge 1/2 - δ$, for any boolean function $f$ holds $I\Big(f(X);Y\Big) \le 1 - H(ε)$.

cs.IT↗

arXiv2015

The "Most informative boolean function" conjecture holds for high noise

We prove the "Most informative boolean function" conjecture of Courtade and Kumar for high noise $ε\ge 1/2 - δ$, for some absolute constant $δ> 0$. Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $ε$, then, provided $ε\ge 1/2 - δ$, for any boolean function $f$ holds $I(f(X);Y) \le 1 - H(ε)$. This conjecture was previously known to hold only for balanced functions.

cs.IT↗