arXiv subjects

Jithin Ravi

Publications and source records attributed to Jithin Ravi.

14 recordsLinked to original sources

arXiv2026

Second-Order Asymptotics of Two-Sample Tests

In two-sampling testing, one observes two independent sequences of independent and identically distributed random variables distributed according to the distributions $P_1$ and $P_2$ and wishes to decide whether $P_1=P_2$ (null hypothesis) or $P_1\neq P_2$ (alternative hypothesis). The Gutman test for this problem compares the empirical distributions of the observed sequences and decides on the null hypothesis if the Jensen-Shannon (JS) divergence between these empirical distributions is below a given threshold. This paper proposes a generalization of the Gutman test, termed \emph{divergence test}, which replaces the JS divergence by an arbitrary divergence. For this test, the exponential decay of the type-II error probability for a fixed type-I error probability is studied. First, it is shown that the divergence test achieves the optimal first-order exponent, irrespective of the choice of divergence. Second, it is demonstrated that divergence tests with invariant divergences achieve the same second-order asymptotics as the Gutman test. In addition, a connection between two-sample testing and robust goodness-of-fit testing is established.

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arXiv2024

On the Second-Order Asymptotics of the Hoeffding Test and Other Divergence Tests

Consider a binary statistical hypothesis testing problem, where $n$ independent and identically distributed random variables $Z^n$ are either distributed according to the null hypothesis $P$ or the alternative hypothesis $Q$, and only $P$ is known. A well-known test that is suitable for this case is the so-called Hoeffding test, which accepts $P$ if the Kullback-Leibler (KL) divergence between the empirical distribution of $Z^n$ and $P$ is below some threshold. This work characterizes the first and second-order terms of the type-II error probability for a fixed type-I error probability for the Hoeffding test as well as for divergence tests, where the KL divergence is replaced by a general divergence. It is demonstrated that, irrespective of the divergence, divergence tests achieve the first-order term of the Neyman-Pearson test, which is the optimal test when both $P$ and $Q$ are known. In contrast, the second-order term of divergence tests is strictly worse than that of the Neyman-Pearson test. It is further demonstrated that divergence tests with an invariant divergence achieve the same second-order term as the Hoeffding test, but divergence tests with a non-invariant divergence may outperform the Hoeffding test for some alternative hypotheses $Q$. Potentially, this behavior could be exploited by a composite hypothesis test with partial knowledge of the alternative hypothesis $Q$ by tailoring the divergence of the divergence test to the set of possible alternative hypotheses.

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arXiv2022

Second-Order Asymptotics of Hoeffding-Like Hypothesis Tests

We consider a binary statistical hypothesis testing problem, where $n$ independent and identically distributed random variables $Z^n$ are either distributed according to the null hypothesis $P$ or the alternate hypothesis $Q$, and only $P$ is known. For this problem, a well-known test is the Hoeffding test, which accepts $P$ if the Kullback-Leibler (KL) divergence between the empirical distribution of $Z^n$ and $P$ is below some threshold. In this paper, we consider Hoeffding-like tests, where the KL divergence is replaced by other divergences, and characterize, for a large class of divergences, the first and second-order terms of the type-II error for a fixed type-I error. Since the considered class includes the KL divergence, we obtain the second-order term of the Hoeffiding test as a special case.

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arXiv2021

Scaling Laws for Gaussian Random Many-Access Channels

This paper considers a Gaussian multiple-access channel with random user activity where the total number of users $\ell_n$ and the average number of active users $k_n$ may grow with the blocklength $n$. For this channel, it studies the maximum number of bits that can be transmitted reliably per unit-energy as a function of $\ell_n$ and $k_n$. When all users are active with probability one, i.e., $\ell_n = k_n$, it is demonstrated that if $k_n$ is of an order strictly below $n/\log n$, then each user can achieve the single-user capacity per unit-energy $(\log e)/N_0$ (where $N_0/ 2$ is the noise power) by using an orthogonal-access scheme. In contrast, if $k_n$ is of an order strictly above $n/\log n$, then the capacity per unit-energy is zero. Consequently, there is a sharp transition between orders of growth where interference-free communication is feasible and orders of growth where reliable communication at a positive rate per unit-energy is infeasible. It is further demonstrated that orthogonal-access schemes in combination with orthogonal codebooks, which achieve the capacity per unit-energy when the number of users is bounded, can be strictly suboptimal. When the user activity is random, i.e., when $\ell_n$ and $k_n$ are different, it is demonstrated that if $k_n\log \ell_n$ is sublinear in $n$, then each user can achieve the single-user capacity per unit-energy $(\log e)/N_0$. Conversely, if $k_n\log \ell_n$ is superlinear in $n$, then the capacity per unit-energy is zero. Consequently, there is again a sharp transition between orders of growth where interference-free communication is feasible and orders of growth where reliable communication at a positive rate is infeasible that depends on the asymptotic behaviours of both $\ell_n$ and $k_n$. It is further demonstrated that orthogonal-access schemes, which are optimal when $\ell_n = k_n$, can be strictly suboptimal.

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arXiv2021

Fundamental Limits of Demand-Private Coded Caching

We consider the coded caching problem with an additional privacy constraint that a user should not get any information about the demands of the other users. We first show that a demand-private scheme for $N$ files and $K$ users can be obtained from a non-private scheme that serves only a subset of the demands for the $N$ files and $NK$ users problem. We further use this fact to construct a demand-private scheme for $N$ files and $K$ users from a particular known non-private scheme for $N$ files and $NK-K+1$ users. It is then demonstrated that, the memory-rate pair $(M,\min \{N,K\}(1-M/N))$, which is achievable for non-private schemes with uncoded transmissions, is also achievable under demand privacy. We further propose a scheme that improves on these ideas by removing some redundant transmissions. The memory-rate trade-off achieved using our schemes is shown to be within a multiplicative factor of 3 from the optimal when $K < N$ and of 8 when $N\leq K$. Finally, we give the exact memory-rate trade-off for demand-private coded caching problems with $N\geq K=2$.

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arXiv2020

Capacity per Unit-Energy of Gaussian Random Many-Access Channels

We consider a Gaussian multiple-access channel with random user activity where the total number of users $\ell_n$ and the average number of active users $k_n$ may be unbounded. For this channel, we characterize the maximum number of bits that can be transmitted reliably per unit-energy in terms of $\ell_n$ and $k_n$. We show that if $k_n\log \ell_n$ is sublinear in $n$, then each user can achieve the single-user capacity per unit-energy. Conversely, if $k_n\log \ell_n$ is superlinear in $n$, then the capacity per unit-energy is zero. We further demonstrate that orthogonal-access schemes, which are optimal when all users are active with probability one, can be strictly suboptimal.

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arXiv2020

Private Index Coding

We study the fundamental problem of index coding under an additional privacy constraint that requires each receiver to learn nothing more about the collection of messages beyond its demanded messages from the server and what is available to it as side information. To enable such private communication, we allow the use of a collection of independent secret keys, each of which is shared amongst a subset of users and is known to the server. The goal is to study properties of the key access structures which make the problem feasible and then design encoding and decoding schemes efficient in the size of the server transmission as well as the sizes of the secret keys. We call this the private index coding problem. We begin by characterizing the key access structures that make private index coding feasible. We also give conditions to check if a given linear scheme is a valid private index code. For up to three users, we characterize the rate region of feasible server transmission and key rates, and show that all feasible rates can be achieved using scalar linear coding and time sharing; we also show that scalar linear codes are sub-optimal for four receivers. The outer bounds used in the case of three users are extended to arbitrary number of users and seen as a generalized version of the well-known polymatroidal bounds for the standard non-private index coding. We also show that the presence of common randomness and private randomness does not change the rate region. Furthermore, we study the case where no keys are shared among the users and provide some necessary and sufficient conditions for feasibility in this setting under a weaker notion of privacy. If the server has the ability to multicast to any subset of users, we demonstrate how this flexibility can be used to provide privacy and characterize the minimum number of server multicasts required.

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arXiv2020

Interactive Secure Function Computation

We consider interactive computation of randomized functions between two users with the following privacy requirement: the interaction should not reveal to either user any extra information about the other user's input and output other than what can be inferred from the user's own input and output. We also consider the case where privacy is required against only one of the users. For both cases, we give single-letter expressions for feasibility and optimal rates of communication. Then we discuss the role of common randomness and interaction in both privacy settings. We also study perfectly secure non-interactive computation when only one of the users computes a randomized function based on a single transmission from the other user. We characterize randomized functions which can be perfectly securely computed in this model and obtain tight bounds on the optimal message lengths in all the privacy settings.

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arXiv2020

Demand-Private Coded Caching and the Exact Trade-off for N=K=2

The distributed coded caching problem has been studied extensively in the recent past. While the known coded caching schemes achieve an improved transmission rate, they violate the privacy of the users since in these schemes the demand of one user is revealed to others in the delivery phase. In this paper, we consider the coded caching problem under the constraint that the demands of the other users remain information theoretically secret from each user. We first show that the memory-rate pair $(M,\min \{N,K\}(1-M/N))$ is achievable under information theoretic demand privacy, while using broadcast transmissions. We then show that a demand-private scheme for $N$ files and $K$ users can be obtained from a non-private scheme that satisfies only a restricted subset of demands of $NK$ users for $N$ files. We then focus on the demand-private coded caching problem for $K=2$ users, $N=2$ files. We characterize the exact memory-rate trade-off for this case. To show the achievability, we use our first result to construct a demand-private scheme from a non-private scheme satisfying a restricted demand subset that is known from an earlier work by Tian. Further, by giving a converse based on the extra requirement of privacy, we show that the obtained achievable region is the exact memory-rate trade-off.

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arXiv2019

Capacity per Unit-Energy of Gaussian Many-Access Channels

We consider a Gaussian multiple-access channel where the number of transmitters grows with the blocklength $n$. For this setup, the maximum number of bits that can be transmitted reliably per unit-energy is analyzed. We show that if the number of users is of an order strictly above $n/\log n$, then the users cannot achieve any positive rate per unit-energy. In contrast, if the number of users is of order strictly below $n/\log n$, then each user can achieve the single-user capacity per unit-energy $(\log e)/N_0$ (where $N_0/ 2$ is the noise power) by using an orthogonal access scheme such as time division multiple access. We further demonstrate that orthogonal codebooks, which achieve the capacity per unit-energy when the number of users is bounded, can be strictly suboptimal.

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arXiv2018

Function Computation through a Bidirectional Relay

We consider a function computation problem in a three node wireless network. Nodes A and B observe two correlated sources $X$ and $Y$ respectively, and want to compute a function $f(X,Y)$. To achieve this, nodes A and B send messages to a relay node C at rates $R_A$ and $R_B$ respectively. The relay C then broadcasts a message to A and B at rate $R_C$. We allow block coding, and study the achievable region of rate triples under both zero-error and $ε$-error. As a preparation, we first consider a broadcast network from the relay to A and B. A and B have side information $X$ and $Y$ respectively. The relay node C observes both $X$ and $Y$ and broadcasts an encoded message to A and B. We want to obtain the optimal broadcast rate such that A and B can recover the function $f(X,Y)$ from the received message and their individual side information $X$ and $Y$ respectively. For this problem, we show equivalence between $ε$-error and zero-error computations-- this gives a rate characterization for zero-error computation. As a corollary, this also gives a rate characterization for the relay network under zero-error for a class of functions called {\em component-wise one-to-one functions} when the support set of $p_{XY}$ is full. For the relay network, the zero-error rate region for arbitrary functions is characterized in terms of graph coloring of some suitably defined probabilistic graphs. We then give a single-letter inner bound to this rate region. Further, we extend the graph theoretic ideas to address the $ε$-error problem and obtain a single-letter inner bound.

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arXiv2016

Broadcast Function Computation with Complementary Side Information

We consider the function computation problem in a three node network with one encoder and two decoders. The encoder has access to two correlated sources $X$ and $Y$. The encoder encodes $X^n$ and $Y^n$ into a message which is given to two decoders. Decoder 1 and decoder 2 have access to $X$ and $Y$ respectively, and they want to compute two functions $f(X,Y)$ and $g(X,Y)$ respectively using the encoded message and their respective side information. We want to find the optimum (minimum) encoding rate under the zero error and $ε$-error (i.e. vanishing error) criteria. For the special case of this problem with $f(X,Y) = Y$ and $g(X,Y) = X$, we show that the $ε$-error optimum rate is also achievable with zero error. This result extends to a more general `complementary delivery index coding' problem with arbitrary number of messages and decoders. For other functions, we show that the cut-set bound is achievable under $ε$-error if $X$ and $Y$ are binary, or if the functions are from a special class of `compatible' functions which includes the case $f=g$.

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arXiv2015

Oblivious Transfer over Wireless Channels

We consider the problem of oblivious transfer (OT) over OFDM and MIMO wireless communication systems where only the receiver knows the channel state information. The sender and receiver also have unlimited access to a noise-free real channel. Using a physical layer approach, based on the properties of the noisy fading channel, we propose a scheme that enables the transmitter to send obliviously one-of-two files, i.e., without knowing which one has been actually requested by the receiver, while also ensuring that the receiver does not get any information about the other file.

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arXiv2014

Zero-Error Function Computation through a Bidirectional Relay

We consider zero error function computation in a three node wireless network. Nodes A and B observe $X$ and $Y$ respectively, and want to compute a function $f(X,Y)$ with zero error. To achieve this, nodes A and B send messages to a relay node C at rates $R_A$ and $R_B$ respectively. The relay C then broadcasts a message to A and B at rate $R_C$ to help them compute $f(X,Y)$ with zero error. We allow block coding, and study the region of rate-triples $(R_A,R_B,R_C)$ that are feasible. The rate region is characterized in terms of graph coloring of some suitably defined probabilistic graphs. We give single letter inner and outer bounds which meet for some simple examples. We provide a sufficient condition on the joint distribution $p_{XY}$ under which the relay can also compute $f(X,Y)$ if A and B can compute it with zero error.

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