In this paper, we study the problem of quantitative group testing (QGT) and analyze the performance of three models: the noiseless model, the additive Gaussian noise model, and the noisy Z-channel model. For each model, we analyze two algorithmic approaches: a linear estimator based on correlation scores, and a least squares estimator (LSE). We derive upper bounds on the number of tests required for exact recovery with vanishing error probability, and complement these results with information-theoretic lower bounds. In the additive Gaussian noise setting, our lower and upper bounds match in order.

We study the Chernoff-Stein exponent of the following binary hypothesis testing problem: Associated with each hypothesis is a set of channels. A transmitter, without knowledge of the hypothesis, chooses the vector of inputs to the channel. Given the hypothesis, from the set associated with the hypothesis, an adversary chooses channels, one for each element of the input vector. Based on the channel outputs, a detector attempts to distinguish between the hypotheses. We study the Chernoff-Stein exponent for the cases where the transmitter (i) is deterministic, (ii) may privately randomize, and (iii) shares randomness with the detector that is unavailable to the adversary. It turns out that while a memoryless transmission strategy is optimal under shared randomness, it may be strictly suboptimal when the transmitter only has private randomness.

We study communication with consensus over a broadcast channel—the receivers reliably decode the sender’s message when the sender is honest, and their decoder outputs agree even if the sender acts maliciously. We characterize the broadcast channels which permit this Byzantine consensus and determine their capacity. We show that communication with consensus is possible only when the broadcast channel has embedded in it a natural “common channel” whose output both receivers can unambiguously determine from their own channel outputs. Interestingly, in general, the consensus capacity may be larger than the point-to-point capacity of the common channel, i.e., while decoding, the receivers may make use of parts of their output signals on which they may not have consensus provided there are some parts (namely, the common channel output) on which they can agree.

We consider the problem of *exact* recovery of a $k$-sparse binary vector from generalized linear measurements (such as *logistic regression*). We analyze the *linear estimation* algorithm (Plan, Vershynin, Yudovina, 2017), and also show information theoretic lower bounds on the number of required measurements. As a consequence of our results, for noisy one bit quantized linear measurements ($\mathsf{1bCS}$), we obtain a sample complexity of $O((k+\sigma^2)\log{n})$, where $\sigma^2$ is the noise variance. This is shown to be optimal due to the information theoretic lower bound. We also obtain tight sample complexity characterization for logistic regression. Since $\mathsf{1bCS}$ is a strictly harder problem than noisy linear measurements ($\mathsf{SparseLinearReg}$) because of added quantization, the same sample complexity is achievable for $\mathsf{SparseLinearReg}$. While this sample complexity can be obtained via the popular lasso algorithm, linear estimation is computationally more efficient. Our lower bound holds for any set of measurements for $\mathsf{SparseLinearReg}$ (similar bound was known for Gaussian measurement matrices) and is closely matched by the maximum-likelihood upper bound. For $\mathsf{SparseLinearReg}$, it was conjectured in Gamarnik and Zadik, 2017 that there is a statistical-computational gap and the number of measurements should be at least $(2k+\sigma^2)\log{n}$ for efficient algorithms to exist. It is worth noting that our results imply that there is no such statistical-computational gap for $\mathsf{1bCS}$ and logistic regression.

We study the distributed function computation problem with$k$users of which at most$s$may be controlled by an adversary and characterize the set of functions of the sources the decoder can reconstruct robustly in the following sense if the users behave honestly, the function is recovered with high probability (w.h.p.); if they behave adversarially, w.h.p, either one of the adversarial users is identified or the function is recovered with vanishingly small distortion.

We introduce the problem of determining the identity of a byzantine user (internal adversary) in a communication system. We consider a two-user discrete memoryless multiple access channel where either user may deviate from the prescribed behaviour. Since small deviations may be indistinguishable from the effects of channel noise, it might be overly restrictive to attempt to detect all deviations. In our formulation, we only require detecting deviations which impede the decoding of the non-deviating user’s message. When neither user deviates, correct decoding is required. When one user deviates, the decoder must either output a pair of messages of which the message of the non-deviating user is correct or identify the deviating user. The users and the receiver do not share any randomness. The results include a characterization of the set of channels where communication is feasible, and an inner and outer bound on the capacity region. We also show that whenever the rate region has non-empty interior, the capacity region is same as the capacity region under randomized encoding, where each user shares independent randomness with the receiver. We also give an outer bound for this randomized coding capacity region.

We consider sequential hypothesis testing among multiple samples of one among$M$pure quantum states in an equidistant ensemble, i.e., those with identical pair-wise inner products. Each measurement in the sequence is a binary projective measurement that collapses the sample of the state measured at that instant into a linear span of a collection of states from the ensemble or its orthogonal complement. The algorithm adaptively decides if additional samples are needed or sufficient observation has been gathered. We show that our sequential measurement algorithm outperforms the sequential testing (ST) receiver, whose error-probability exponent is known to achieve the quantum Chernoff bound asymptotically in the limit of large (and fixed) number of samples. Even though our algorithm does not attain the quantum limit of minimum error probability (the Helstrom limit), it paves the way for future research on more advanced sequential quantum hypothesis tests, e.g., those that go beyond binary projective measurements on each sample.

We study communication over a Multiple Access Channel (MAC) where users can possibly be adversarial. The receiver is unaware of the identity of the adversarial users (if any). When all users are non-adversarial, we want their messages to be decoded reliably. When a user behaves adversarially, we require that the honest users’ messages be decoded reliably. An adversarial user can mount an attack by sending any input into the channel rather than following the protocol. It turns out that the 2-user MAC capacity region follows from the point-to-point Arbitrarily Varying Channel (AVC) capacity. For the 3-user MAC in which at most one user may be malicious, we characterize the capacity region for deterministic codes and randomized codes (where each user shares an independent random secret key with the receiver). These results are then generalized for the$k$-user MAC where the adversary may control all users in one out of a collection of given subsets.

We study the Chernoff-Stein exponent of the following binary hypothesis testing problem: Associated with each hypothesis is a set of channels. A transmitter, without knowledge of the hypothesis, chooses the vector of inputs to the channel. Given the hypothesis, from the set associated with the hypothesis, an adversary chooses channels, one for each element of the input vector. Based on the channel outputs, a detector attempts to distinguish between the hypotheses. We study the Chernoff-Stein exponent for the cases where the transmitter (i) is deterministic, (ii) may privately randomize, and (iii) shares randomness with the detector that is unavailable to the adversary. It turns out that while a memoryless transmission strategy is optimal under shared randomness, it may be strictly suboptimal when the transmitter only has private randomness.

We study communication with consensus over a broadcast channel - the receivers reliably decode the sender’s message when the sender is honest, and their decoder outputs agree even if the sender acts maliciously. We characterize the broadcast channels which permit this byzantine consensus and determine their capacity.

We propose a communication model, that we call compound arbitrarily varying channels (CAVC), which unifies and generalizes compound channels and arbitrarily varying channels (AVC). A CAVC can be viewed as a noisy channel with a fixed, but unknown, compound-state and an AVC-state which may vary with every channel use. The AVC-state is controlled by an adversary who is aware of the compound-state. We study three problems in this setting: ‘communication’, ‘communication and compound-state identification’, and ‘communication or compound-state identification’. For these problems, we study conditions for feasibility and capacity under deterministic coding and random coding.

We introduce the problem of determining the identity of a byzantine user (internal adversary) in a communication system. We consider a two-user discrete memoryless multiple access channel where either user may deviate from the prescribed behaviour. Owing to the noisy nature of the channel, it may be overly restrictive to attempt to detect all deviations. In our formulation, we only require detecting deviations which impede the decoding of the non-deviating user's message. When neither user deviates, correct decoding is required. When one user deviates, the decoder must either output a pair of messages of which the message of the non-deviating user is correct or identify the deviating user. The users and the receiver do not share any randomness. The results include a characterization of the set of channels where communication is feasible, and an inner and outer bound on the capacity region.

We study authenticated communication over two-user multiple access channels (MAC) where one of the users is possibly adversarial. When both users behave non-adversarially, we want their messages to be decoded reliably. However, we also want to ensure that an adversarial user cannot cause an undetected error on the other (honest) user's message. We show that the following three-phase scheme is rate-optimal: a standard MAC code is first used to achieve unauthenticated communication; this is followed by two authentication phases where each user authenticates their message treating the other user as a possible adversary. We show that the authentication phases can be very short since this form of authentication itself, when possible, can be achieved for message sets whose size grow doubly exponentially in blocklength. This leads to our result that the authenticated communication capacity region of a discrete memoryless MAC is either zero or the (unauthenticated) MAC capacity region itself. This also, arguably, explains the similar nature of authenticated communication capacity of a discrete memoryless point-to-point adversarial channel recently found by Kosut and Kliewer (ITW, 2018). We also obtain analogous results for additive Gaussian noise channels.

Communication over a three-user multiple access channel (MAC) is studied when any one of the users may behave adversarially. The capacity region is characterized for randomized codes (where each user shares an independent secret key with the receiver). The capacity region for deterministic codes is also studied. Necessary conditions including a new non-symmetrizability condition is obtained for this capacity region to be non-trivial. It is shown that when none of the users are symmetrizable, the randomized coding capacity region is also achievable with deterministic codes. This is analogous to the result of Ahlswede and Cai (1991) for arbitrarily varying MAC.