arXiv subjects

Shashank Vatedka

Publications and source records attributed to Shashank Vatedka.

17 recordsLinked to original sources

arXiv2025

Compression with Privacy-Preserving Random Access

We show that an i.i.d.\ binary source sequence $X_1,\ldots,X_n$ can be losslessly compressed at any rate above entropy while ensuring that the decoding of any $X_i$ reveals no information about the remaining symbols $\{X_j : j \neq i\}$. This problem reduces to a marginal consistency problem induced by the simultaneous privacy and reliability constraints. To address it, we develop a technique based on a geometric representation of codeword distributions, which may be of independent interest.

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arXiv2025

One-Bit Distributed Mean Estimation with Unknown Variance

In this work, we study the problem of distributed mean estimation with 1-bit communication constraints when the variance is unknown. We focus on the setting where each user has access to one iid sample drawn from a distribution belonging to a location-scale family, and is limited to sending just a single bit of information to a central server whose goal is to estimate the mean. We propose simple non-adaptive and adaptive protocols and show that both achieve asymptotic normality. We derive bounds on the asymptotic (in the number of users) Mean Squared Error (MSE) achieved by these protocols. For a class of symmetric log-concave distributions, we derive matching lower bounds for the MSE of adaptive protocols, establishing the optimality of our scheme. Furthermore, we develop a lower bound on the MSE for non-adaptive protocols that applies to any symmetric strictly log-concave distribution, using a refined squared Hellinger distance analysis. Through this, we show that for many common distributions, including a subclass of the generalized Gaussian family, the asymptotic minimax MSE achieved by the best non-adaptive protocol is strictly larger than that achieved by our simple adaptive protocol. We also demonstrate that increasing the number of bits per user can only marginally reduce the asymptotic MSE of adaptive protocols. Our simulation results confirm a positive gap between the adaptive and non-adaptive settings, aligning with the theoretical bounds.

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arXiv2022

Bounds for Multiple Packing and List-Decoding Error Exponents

We revisit the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set $\mathcal{C}$ of points in $ \mathbb{R}^n $ such that any point in $ \mathbb{R}^n $ lies in the intersection of at most $ L-1 $ balls of radius $ \sqrt{nN} $ around points in $ \mathcal{C} $. We study the multiple packing problem for both bounded point sets whose points have norm at most $\sqrt{nP}$ for some constant $P>0$ and unbounded point sets whose points are allowed to be anywhere in $ \mathbb{R}^n $. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive various bounds on the largest possible density of a multiple packing in both bounded and unbounded settings. A related notion called average-radius multiple packing is also studied. Some of our lower bounds exactly pin down the asymptotics of certain ensembles of average-radius list-decodable codes, e.g., (expurgated) Gaussian codes and (expurgated) Poisson Point Processes. To this end, we apply tools from high-dimensional geometry and large deviation theory. Some of our lower bounds on the optimal multiple packing density are the best known lower bounds. These bounds are obtained via a proxy known as error exponent. The latter quantity is the best exponent of the probability of list-decoding error when the code is corrupted by a Gaussian noise. We establish a curious inequality which relates the error exponent, a quantity of average-case nature, to the list-decoding radius, a quantity of worst-case nature. We derive various bounds on the error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.

math.MG↗

arXiv2022

Multiple Packing: Lower and Upper Bounds

We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set $\mathcal{C}$ of points in $ \mathbb{R}^n $ such that any point in $ \mathbb{R}^n $ lies in the intersection of at most $ L-1 $ balls of radius $ \sqrt{nN} $ around points in $ \mathcal{C} $. We study the multiple packing problem for both bounded point sets whose points have norm at most $\sqrt{nP}$ for some constant $P>0$ and unbounded point sets whose points are allowed to be anywhere in $ \mathbb{R}^n $. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive various bounds on the largest possible density of a multiple packing in both bounded and unbounded settings. A related notion called average-radius multiple packing is also studied. Some of our lower bounds exactly pin down the asymptotics of certain ensembles of average-radius list-decodable codes, e.g., (expurgated) Gaussian codes and (expurgated) spherical codes. In particular, our lower bound obtained from spherical codes is the best known lower bound on the optimal multiple packing density and is the first lower bound that approaches the known large $L$ limit under the average-radius notion of multiple packing. To derive these results, we apply tools from high-dimensional geometry and large deviation theory.

math.MG↗

arXiv2022

Multiple Packing: Lower Bounds via Infinite Constellations

math.MG↗

arXiv2022

Multiple Packing: Lower Bounds via Error Exponents

We derive lower bounds on the maximal rates for multiple packings in high-dimensional Euclidean spaces. Multiple packing is a natural generalization of the sphere packing problem. For any $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $, a multiple packing is a set $\mathcal{C}$ of points in $ \mathbb{R}^n $ such that any point in $ \mathbb{R}^n $ lies in the intersection of at most $ L-1 $ balls of radius $ \sqrt{nN} $ around points in $ \mathcal{C} $. We study this problem for both bounded point sets whose points have norm at most $\sqrt{nP}$ for some constant $P>0$ and unbounded point sets whose points are allowed to be anywhere in $ \mathbb{R}^n $. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. We derive the best known lower bounds on the optimal multiple packing density. This is accomplished by establishing a curious inequality which relates the list-decoding error exponent for additive white Gaussian noise channels, a quantity of average-case nature, to the list-decoding radius, a quantity of worst-case nature. We also derive various bounds on the list-decoding error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.

math.MG↗

arXiv2022

Local Decoding in Distributed Compression

It was recently shown that the lossless compression of a single source $X^n$ is achievable with a notion of strong locality; any $X_i$ can be decoded from a constant number of compressed bits, with a vanishing in $n$ probability of error. By contrast, we show that for two separately encoded sources $(X^n,Y^n)$, lossless compression and strong locality is generally not possible. Specifically, we show that for the class of ``confusable'' sources, strong locality cannot be achieved whenever one of the sources is compressed below its entropy. Irrespective of $n$, for some index $i$ the probability of error of decoding $(X_i,Y_i)$ is lower bounded by $2^{-O(d)}$, where $d$ denotes the number of compressed bits accessed by the local decoder. Conversely, if the source is not confusable, strong locality is possible even if one of the sources is compressed below its entropy. Results extend to an arbitrary number of sources.

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arXiv2021

List Decoding Random Euclidean Codes and Infinite Constellations

We study the list decodability of different ensembles of codes over the real alphabet under the assumption of an omniscient adversary. It is a well-known result that when the source and the adversary have power constraints $ P $ and $ N $ respectively, the list decoding capacity is equal to $ \frac{1}{2}\log\frac{P}{N} $. Random spherical codes achieve constant list sizes, and the goal of the present paper is to obtain a better understanding of the smallest achievable list size as a function of the gap to capacity. We show a reduction from arbitrary codes to spherical codes, and derive a lower bound on the list size of typical random spherical codes. We also give an upper bound on the list size achievable using nested Construction-A lattices and infinite Construction-A lattices. We then define and study a class of infinite constellations that generalize Construction-A lattices and prove upper and lower bounds for the same. Other goodness properties such as packing goodness and AWGN goodness of infinite constellations are proved along the way. Finally, we consider random lattices sampled from the Haar distribution and show that if a certain number-theoretic conjecture is true, then the list size grows as a polynomial function of the gap-to-capacity.

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arXiv2020

Quadratically Constrained Myopic Adversarial Channels

We study communication in the presence of a jamming adversary where quadratic power constraints are imposed on the transmitter and the jammer. The jamming signal is allowed to be a function of the codebook, and a noncausal but noisy observation of the transmitted codeword. For a certain range of the noise-to-signal ratios (NSRs) of the transmitter and the jammer, we are able to characterize the capacity of this channel under deterministic encoding or stochastic encoding, i.e., with no common randomness between the encoder/decoder pair. For the remaining NSR regimes, we determine the capacity under the assumption of a small amount of common randomness (at most $2\log(n)$ bits in one sub-regime, and at most $Ω(n)$ bits in the other sub-regime) available to the encoder-decoder pair. Our proof techniques involve a novel myopic list-decoding result for achievability, and a Plotkin-type push attack for the converse in a subregion of the NSRs, both of which which may be of independent interest. We also give bounds on the secrecy capacity of this channel assuming that the jammer is simultaneously eavesdropping.

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arXiv2020

$O(\log \log n)$ Worst-Case Local Decoding and Update Efficiency for Data Compression

This paper addresses the problem of data compression with local decoding and local update. A compression scheme has worst-case local decoding $d_{wc}$ if any bit of the raw file can be recovered by probing at most $d_{wc}$ bits of the compressed sequence, and has update efficiency of $u_{wc}$ if a single bit of the raw file can be updated by modifying at most $u_{wc}$ bits of the compressed sequence. This article provides an entropy-achieving compression scheme for memoryless sources that simultaneously achieves $ O(\log\log n) $ local decoding and update efficiency. Key to this achievability result is a novel succinct data structure for sparse sequences which allows efficient local decoding and local update. Under general assumptions on the local decoder and update algorithms, a converse result shows that $d_{wc}$ and $u_{wc}$ must grow as $ Ω(\log\log n) $.

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arXiv2020

Quadratically Constrained Two-way Adversarial Channels

We study achievable rates of reliable communication in a power-constrained two-way additive interference channel over the real alphabet where communication is disrupted by a power-constrained jammer. This models the wireless communication scenario where two users Alice and Bob, operating in the full duplex mode, wish to exchange messages with each other in the presence of a jammer, James. Alice and Bob simultaneously transmit their encodings $ \underline{x}_A $ and $\underline{x}_B $ over $ n $ channel uses. It is assumed that James can choose his jamming signal $ \underline{s} $ as a noncausal randomized function of $ \underline{x}_A $ and $ \underline{x}_B $, and the codebooks used by Alice and Bob. Alice and Bob observe $ \underline{x}_A+\underline{x}_B +\underline{s}$, and must recover each others' messages reliably. In this article, we provide upper and lower bounds on the capacity of this channel which match each other and equal $ \frac{1}{2}\log\left(\frac{1}{2} + \mathsf{SNR}\right) $ in the high-$\mathsf{SNR}$ regime (where $\mathsf{SNR}$, signal to noise ratios, is defined as the ratio of the power constraints of the users to the power constraint of the jammer). We give a code construction based on lattice codes, and derive achievable rates for large $\mathsf{SNR}$. We also present upper bounds based on two specific attack strategies for James. Along the way, sumset property of lattices for the achievability and general properties of capacity-achieving codes for memoryless channels for the converse are proved, which might be of independent interest.

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arXiv2019

Local Decode and Update for Big Data Compression

This paper investigates data compression that simultaneously allows local decoding and local update. The main result is a universal compression scheme for memoryless sources with the following features. The rate can be made arbitrarily close to the entropy of the underlying source, contiguous fragments of the source can be recovered or updated by probing or modifying a number of codeword bits that is on average linear in the size of the fragment, and the overall encoding and decoding complexity is quasilinear in the blocklength of the source. In particular, the local decoding or update of a single message symbol can be performed by probing or modifying a constant number of codeword bits. This latter part improves over previous best known results for which local decodability or update efficiency grows logarithmically with blocklength.

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arXiv2016

Improving the Performance of Nested Lattice Codes Using Concatenation

A fundamental problem in coding theory is the design of an efficient coding scheme that achieves the capacity of the additive white Gaussian (AWGN) channel. The main objective of this short note is to point out that by concatenating a capacity-achieving nested lattice code with a suitable high-rate linear code over an appropriate finite field, we can achieve the capacity of the AWGN channel with polynomial encoding and decoding complexity. Specifically, we show that using inner Construction-A lattice codes and outer Reed-Solomon codes, we can obtain capacity-achieving codes whose encoding and decoding complexities grow as $O(N^2)$, while the probability of error decays exponentially in $N$, where $N$ denotes the blocklength. Replacing the outer Reed-Solomon code by an expander code helps us further reduce the decoding complexity to $O(N\log^2N)$. This also gives us a recipe for converting a high-complexity nested lattice code for a Gaussian channel to a low-complexity concatenated code without any loss in the asymptotic rate. As examples, we describe polynomial-time coding schemes for the wiretap channel, and the compute-and-forward scheme for computing integer linear combinations of messages.

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arXiv2016

Some "Goodness" Properties of LDA Lattices

We study some structural properties of Construction-A lattices obtained from low density parity check (LDPC) codes over prime fields. Such lattices are called low density Construction-A (LDA) lattices, and permit low-complexity belief propagation decoding for transmission over Gaussian channels. It has been shown that LDA lattices achieve the capacity of the power constrained additive white Gaussian noise (AWGN) channel with closest lattice-point decoding, and simulations suggested that they perform well under belief propagation decoding. We continue this line of work, and prove that these lattices are good for packing and mean squared error (MSE) quantization, and that their duals are good for packing. With this, we can conclude that codes constructed using nested LDA lattices can achieve the capacity of the power constrained AWGN channel, the capacity of the dirty paper channel, the rates guaranteed by the compute-and-forward protocol, and the best known rates for bidirectional relaying with perfect secrecy.

cs.IT↗

arXiv2016

A Lattice Coding Scheme for Secret Key Generation from Gaussian Markov Tree Sources

In this article, we study the problem of secret key generation in the multiterminal source model, where the terminals have access to correlated Gaussian sources. We assume that the sources form a Markov chain on a tree. We give a nested lattice-based key generation scheme whose computational complexity is polynomial in the number, N , of independent and identically distributed samples observed by each source. We also compute the achievable secret key rate and give a class of examples where our scheme is optimal in the fine quantization limit. However, we also give examples that show that our scheme is not always optimal in the limit of fine quantization.

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arXiv2015

Nested Lattice Codes for Secure Bidirectional Relaying with Asymmetric Channel Gains

The basic problem of secure bidirectional relaying involves two users who want to exchange messages via an intermediate "honest-but-curious" relay node. There is no direct link between the users, all communication must take place via the relay node. The links between the user nodes and the relay are wireless links with Gaussian noise. It is required that the users' messages be kept secure from the relay. In prior work, we proposed coding schemes based on nested lattices for this problem, assuming that the channel gains from the two user nodes to the relay are identical. We also analyzed the power-rate tradeoff for secure and reliable message exchange using our coding schemes. In this paper, we extend our prior work to the case when the channel gains are not necessarily identical, and are known to the relay node but perhaps not to the users. We show that using our scheme, perfect secrecy can be obtained only for certain values of the channel gains, and analyze the power-rate tradeoff in these cases. We also make similar observations for our strongly-secure scheme.

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arXiv2014

Secure Compute-and-Forward in a Bidirectional Relay

We consider the basic bidirectional relaying problem, in which two users in a wireless network wish to exchange messages through an intermediate relay node. In the compute-and-forward strategy, the relay computes a function of the two messages using the naturally-occurring sum of symbols simultaneously transmitted by user nodes in a Gaussian multiple access (MAC) channel, and the computed function value is forwarded to the user nodes in an ensuing broadcast phase. In this paper, we study the problem under an additional security constraint, which requires that each user's message be kept secure from the relay. We consider two types of security constraints: perfect secrecy, in which the MAC channel output seen by the relay is independent of each user's message; and strong secrecy, which is a form of asymptotic independence. We propose a coding scheme based on nested lattices, the main feature of which is that given a pair of nested lattices that satisfy certain "goodness" properties, we can explicitly specify probability distributions for randomization at the encoders to achieve the desired security criteria. In particular, our coding scheme guarantees perfect or strong secrecy even in the absence of channel noise. The noise in the channel only affects reliability of computation at the relay, and for Gaussian noise, we derive achievable rates for reliable and secure computation. We also present an application of our methods to the multi-hop line network in which a source needs to transmit messages to a destination through a series of intermediate relays.

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