arXiv subjects

Nir Weinberger

Publications and source records attributed to Nir Weinberger.

At least 55 records · Page 3Linked to original sources

arXiv2022

Error Probability Bounds for Coded-Index DNA Storage

The DNA storage channel is considered, in which a codeword is comprised of $M$ unordered DNA molecules. At reading time, $N$ molecules are sampled with replacement, and then each molecule is sequenced. A coded-index concatenated-coding scheme is considered, in which the $m$th molecule of the codeword is restricted to a subset of all possible molecules (an inner code), which is unique for each $m$. The decoder has low-complexity, and is based on first decoding each molecule separately (the inner code), and then decoding the sequence of molecules (an outer code). Only mild assumptions are made on the sequencing channel, in the form of the existence of an inner code and decoder with vanishing error. The error probability of a random code as well as an expurgated code is analyzed and shown to decay exponentially with $N$. This establishes the importance of increasing the coverage depth $N/M$ in order to obtain low error probability.

cs.IT↗

arXiv2022

The Compound Information Bottleneck Outlook

We formulate and analyze the compound information bottleneck programming. In this problem, a Markov chain $ \mathsf{X} \rightarrow \mathsf{Y} \rightarrow \mathsf{Z} $ is assumed with fixed marginal distributions $\mathsf{P}_{\mathsf{X}}$ and $\mathsf{P}_{\mathsf{Y}}$, and the mutual information between $ \mathsf{X} $ and $ \mathsf{Z} $ is sought to be maximized over the choice of conditional probability of $\mathsf{Z}$ given $\mathsf{Y}$ from a given class, under the \textit{worst choice} of the joint probability of the pair $(\mathsf{X},\mathsf{Y})$ from a different class. We consider several classes based on extremes of: mutual information; minimal correlation; total variation; and the relative entropy class. We provide values, bounds, and various characterizations for specific instances of this problem: the binary symmetric case, the scalar Gaussian case, the vector Gaussian case and the symmetric modulo-additive case. Finally, for the general case, we propose a Blahut-Arimoto type of alternating iterations algorithm to find a consistent solution to this problem.

cs.IT↗

arXiv2022

The DNA Storage Channel: Capacity and Error Probability

The DNA storage channel is considered, in which the $M$ Deoxyribonucleic acid (DNA) molecules comprising each codeword are stored without order, sampled $N$ times with replacement, and then sequenced over a discrete memoryless channel. For a constant coverage depth $M/N$ and molecule length scaling $Θ(\log M)$, lower (achievability) and upper (converse) bounds on the capacity of the channel, as well as a lower (achievability) bound on the reliability function of the channel are provided. Both the lower and upper bounds on the capacity generalize a bound which was previously known to hold only for the binary symmetric sequencing channel, and only under certain restrictions on the molecule length scaling and the crossover probability parameters. When specified to binary symmetric sequencing channel, these restrictions are completely removed for the lower bound and are significantly relaxed for the upper bound in the high-noise regime. The lower bound on the reliability function is achieved under a universal decoder, and reveals that the dominant error event is that of outage -- the event in which the capacity of the channel induced by the DNA molecule sampling operation does not support the target rate.

cs.IT↗

arXiv2022

Robust Linear Regression for General Feature Distribution

We investigate robust linear regression where data may be contaminated by an oblivious adversary, i.e., an adversary than may know the data distribution but is otherwise oblivious to the realizations of the data samples. This model has been previously analyzed under strong assumptions. Concretely, $\textbf{(i)}$ all previous works assume that the covariance matrix of the features is positive definite; and $\textbf{(ii)}$ most of them assume that the features are centered (i.e. zero mean). Additionally, all previous works make additional restrictive assumption, e.g., assuming that the features are Gaussian or that the corruptions are symmetrically distributed. In this work we go beyond these assumptions and investigate robust regression under a more general set of assumptions: $\textbf{(i)}$ we allow the covariance matrix to be either positive definite or positive semi definite, $\textbf{(ii)}$ we do not necessarily assume that the features are centered, $\textbf{(iii)}$ we make no further assumption beyond boundedness (sub-Gaussianity) of features and measurement noise. Under these assumption we analyze a natural SGD variant for this problem and show that it enjoys a fast convergence rate when the covariance matrix is positive definite. In the positive semi definite case we show that there are two regimes: if the features are centered we can obtain a standard convergence rate; otherwise the adversary can cause any learner to fail arbitrarily.

cs.LG↗

arXiv2021

Generalization Bounds and Algorithms for Learning to Communicate over Additive Noise Channels

An additive noise channel is considered, in which the distribution of the noise is nonparametric and unknown. The problem of learning encoders and decoders based on noise samples is considered. For uncoded communication systems, the problem of choosing a codebook and possibly also a generalized minimal distance decoder (which is parameterized by a covariance matrix) is addressed. High probability generalization bounds for the error probability loss function, as well as for a hinge-type surrogate loss function are provided. A stochastic-gradient based alternating-minimization algorithm for the latter loss function is proposed. In addition, a Gibbs-based algorithm that gradually expurgates an initial codebook from codewords in order to obtain a smaller codebook with improved error probability is proposed, and bounds on its average empirical error and generalization error, as well as a high probability generalization bound, are stated. Various experiments demonstrate the performance of the proposed algorithms. For coded systems, the problem of maximizing the mutual information between the input and the output with respect to the input distribution is addressed, and uniform convergence bounds for two different classes of input distributions are obtained.

cs.IT↗

arXiv2021

The EM Algorithm is Adaptively-Optimal for Unbalanced Symmetric Gaussian Mixtures

This paper studies the problem of estimating the means $\pmθ_{*}\in\mathbb{R}^{d}$ of a symmetric two-component Gaussian mixture $δ_{*}\cdot N(θ_{*},I)+(1-δ_{*})\cdot N(-θ_{*},I)$ where the weights $δ_{*}$ and $1-δ_{*}$ are unequal. Assuming that $δ_{*}$ is known, we show that the population version of the EM algorithm globally converges if the initial estimate has non-negative inner product with the mean of the larger weight component. This can be achieved by the trivial initialization $θ_{0}=0$. For the empirical iteration based on $n$ samples, we show that when initialized at $θ_{0}=0$, the EM algorithm adaptively achieves the minimax error rate $\tilde{O}\Big(\min\Big\{\frac{1}{(1-2δ_{*})}\sqrt{\frac{d}{n}},\frac{1}{\|θ_{*}\|}\sqrt{\frac{d}{n}},\left(\frac{d}{n}\right)^{1/4}\Big\}\Big)$ in no more than $O\Big(\frac{1}{\|θ_{*}\|(1-2δ_{*})}\Big)$ iterations (with high probability). We also consider the EM iteration for estimating the weight $δ_{*}$, assuming a fixed mean $θ$ (which is possibly mismatched to $θ_{*}$). For the empirical iteration of $n$ samples, we show that the minimax error rate $\tilde{O}\Big(\frac{1}{\|θ_{*}\|}\sqrt{\frac{d}{n}}\Big)$ is achieved in no more than $O\Big(\frac{1}{\|θ_{*}\|^{2}}\Big)$ iterations. These results robustify and complement recent results of Wu and Zhou obtained for the equal weights case $δ_{*}=1/2$.

math.ST↗

arXiv2019

Large Deviations Behavior of the Logarithmic Error Probability of Random Codes

This work studies the deviations of the error exponent of the constant composition code ensemble around its expectation, known as the error exponent of the typical random code (TRC). In particular, it is shown that the probability of randomly drawing a codebook whose error exponent is smaller than the TRC exponent is exponentially small; upper and lower bounds for this exponent are given, which coincide in some cases. In addition, the probability of randomly drawing a codebook whose error exponent is larger than the TRC exponent is shown to be double-exponentially small; upper and lower bounds to the double-exponential exponent are given. The results suggest that codebooks whose error exponent is larger than the error exponent of the TRC are extremely rare. The key ingredient in the proofs is a new large deviations result of type class enumerators with dependent variables.

cs.IT↗

arXiv2019

On the Reliability Function of Distributed Hypothesis Testing Under Optimal Detection

The distributed hypothesis testing problem with full side-information is studied. The trade-off (reliability function) between the two types of error exponents under limited rate is studied in the following way. First, the problem is reduced to the problem of determining the reliability function of channel codes designed for detection (in analogy to a similar result which connects the reliability function of distributed lossless compression and ordinary channel codes). Second, a single-letter random-coding bound based on a hierarchical ensemble, as well as a single-letter expurgated bound, are derived for the reliability of channel-detection codes. Both bounds are derived for a system which employs the optimal detection rule. We conjecture that the resulting random-coding bound is ensemble-tight, and consequently optimal within the class of quantization-and-binning schemes.

cs.IT↗

arXiv2019

Self-Predicting Boolean Functions

A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$, if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is obtained from $X^{n}$ by independently flipping each coordinate with probability $δ$. This paper is about self-predicting functions, which are those that coincide with their optimal predictor.

cs.DM↗

arXiv2018

On the VC-Dimension of Binary Codes

We investigate the asymptotic rates of length-$n$ binary codes with VC-dimension at most $dn$ and minimum distance at least $δn$. Two upper bounds are obtained, one as a simple corollary of a result by Haussler and the other via a shortening approach combining Sauer-Shelah lemma and the linear programming bound. Two lower bounds are given using Gilbert-Varshamov type arguments over constant-weight and Markov-type sets.

cs.IT↗

arXiv2018

Guessing with a Bit of Help

What is the value of a single bit to a guesser? We study this problem in a setup where Alice wishes to guess an i.i.d. random vector, and can procure one bit of information from Bob, who observes this vector through a memoryless channel. We are interested in the guessing efficiency, which we define as the best possible multiplicative reduction in Alice's guessing-moments obtainable by observing Bob's bit. For the case of a uniform binary vector observed through a binary symmetric channel, we provide two lower bounds on the guessing efficiency by analyzing the performance of the Dictator and Majority functions, and two upper bounds via maximum entropy and Fourier-analytic / hypercontractivity arguments. We then extend our maximum entropy argument to give a lower bound on the guessing efficiency for a general channel with a binary uniform input, via the strong data-processing inequality constant of the reverse channel. We compute this bound for the binary erasure channel, and conjecture that Greedy Dictator functions achieve the guessing efficiency.

cs.IT↗

arXiv2017

Expurgated Bounds for the Asymmetric Broadcast Channel

This work contains two main contributions concerning the expurgation of hierarchical ensembles for the asymmetric broadcast channel. The first is an analysis of the optimal maximum likelihood (ML) decoders for the weak and strong user. Two different methods of code expurgation will be used, that will provide two competing error exponents. The second is the derivation of expurgated exponents under the generalized stochastic likelihood decoder (GLD). We prove that the GLD exponents are at least as tight as the maximum between the random coding error exponents derived in an earlier work by Averbuch and Merhav (2017) and one of our ML-based expurgated exponents. By that, we actually prove the existence of hierarchical codebooks that achieve the best of the random coding exponent and the expurgated exponent simultaneously for both users.

cs.IT↗

arXiv2017

Erasure/List Random Coding Error Exponents Are Not Universally Achievable

We study the problem of universal decoding for unknown discrete memoryless channels in the presence of erasure/list option at the decoder, in the random coding regime. Specifically, we harness a universal version of Forney's classical erasure/list decoder developed in earlier studies, which is based on the competitive minimax methodology, and guarantees universal achievability of a certain fraction of the optimum random coding error exponents. In this paper, we derive an exact single-letter expression for the maximum achievable fraction. Examples are given in which the maximal achievable fraction is strictly less than unity, which imply that, in general, there is no universal erasure/list decoder which achieves the same random coding error exponents as the optimal decoder for a known channel. This is in contrast to the situation in ordinary decoding (without the erasure/list option), where optimum exponents are universally achievable, as is well known. It is also demonstrated that previous lower bounds derived for the maximal achievable fraction are not tight in general. We then analyze a generalized random coding ensemble which incorporate a training sequence, in conjunction with a suboptimal practical decoder ("plug-in" decoder), which first estimates the channel using the known training sequence, and then decodes the remaining symbols of the codeword using the estimated channel. One of the implications of our results, is setting the stage for a reasonable criterion of optimal training. Finally, we compare the performance of the "plug-in" decoder and the universal decoder, in terms of the achievable error exponents, and show that the latter is noticeably better than the former.

cs.IT↗

arXiv2016

On the Optimal Boolean Function for Prediction under Quadratic Loss

Suppose $Y^{n}$ is obtained by observing a uniform Bernoulli random vector $X^{n}$ through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between $Y^{n}$ and a Boolean function $\mathsf{b}(X^{n})$ could be, and conjectured that the maximum is attained by a dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in a sequential prediction of $Y^{n}$ under logarithmic loss, given $\mathsf{b}(X^{n})$. In this paper, we study the question of minimizing the sequential prediction cost under a different (proper) loss function - the quadratic loss. In the noiseless case, we show that majority asymptotically minimizes this prediction cost among all Boolean functions. We further show that for weak noise, majority is better than dictator, and that for strong noise dictator outperforms majority. We conjecture that for quadratic loss, there is no single sequence of Boolean functions that is simultaneously (asymptotically) optimal at all noise levels.

cs.IT↗

arXiv2016

Lower Bounds on Parameter Modulation-Estimation Under Bandwidth Constraints

We consider the problem of modulating the value of a parameter onto a band-limited signal to be transmitted over a continuous-time, additive white Gaussian noise (AWGN) channel, and estimating this parameter at the receiver. The performance is measured by the mean power-$α$ error (MP$α$E), which is defined as the worst-case $α$-th order moment of the absolute estimation error. The optimal exponential decay rate of the MP$α$E as a function of the transmission time, is investigated. Two upper (converse) bounds on the MP$α$E exponent are derived, on the basis of known bounds for the AWGN channel of inputs with unlimited bandwidth. The bounds are computed for typical values of the error moment and the signal-to-noise ratio (SNR), and the SNR asymptotics of the different bounds are analyzed. The new bounds are compared to known converse and achievability bounds, which were derived from channel coding considerations.

cs.IT↗

arXiv2015

Channel Detection in Coded Communication

We consider the problem of block-coded communication, where in each block, the channel law belongs to one of two disjoint sets. The decoder is aimed to decode only messages that have undergone a channel from one of the sets, and thus has to detect the set which contains the prevailing channel. We begin with the simplified case where each of the sets is a singleton. For any given code, we derive the optimum detection/decoding rule in the sense of the best trade-off among the probabilities of decoding error, false alarm, and misdetection, and also introduce sub-optimal detection/decoding rules which are simpler to implement. Then, various achievable bounds on the error exponents are derived, including the exact single-letter characterization of the random coding exponents for the optimal detector/decoder. We then extend the random coding analysis to general sets of channels, and show that there exists a universal detector/decoder which performs asymptotically as well as the optimal detector/decoder, when tuned to detect a channel from a specific pair of channels. The case of a pair of binary symmetric channels is discussed in detail.

cs.IT↗

arXiv2015

A Large Deviations Approach to Secure Lossy Compression

We consider a Shannon cipher system for memoryless sources, in which distortion is allowed at the legitimate decoder. The source is compressed using a rate distortion code secured by a shared key, which satisfies a constraint on the compression rate, as well as a constraint on the exponential rate of the excess-distortion probability at the legitimate decoder. Secrecy is measured by the exponential rate of the exiguous-distortion probability at the eavesdropper, rather than by the traditional measure of equivocation. We define the perfect secrecy exponent as the maximal exiguous-distortion exponent achievable when the key rate is unlimited. Under limited key rate, we prove that the maximal achievable exiguous-distortion exponent is equal to the minimum between the average key rate and the perfect secrecy exponent, for a fairly general class of variable key rate codes.

cs.IT↗

arXiv2014

Simplified Erasure/List Decoding

We consider the problem of erasure/list decoding using certain classes of simplified decoders. Specifically, we assume a class of erasure/list decoders, such that a codeword is in the list if its likelihood is larger than a threshold. This class of decoders both approximates the optimal decoder of Forney, and also includes the following simplified subclasses of decoding rules: The first is a function of the output vector only, but not the codebook (which is most suitable for high rates), and the second is a scaled version of the maximum likelihood decoder (which is most suitable for low rates). We provide single-letter expressions for the exact random coding exponents of any decoder in these classes, operating over a discrete memoryless channel. For each class of decoders, we find the optimal decoder within the class, in the sense that it maximizes the erasure/list exponent, under a given constraint on the error exponent. We establish the optimality of the simplified decoders of the first and second kind for low and high rates, respectively.

cs.IT↗