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Mixed-integer convex representable (MICP-R) sets are those sets that can be represented exactly through a mixed-integer convex programming formulation. Following up on recent work by Lubin et al. (2017, 2020) we investigate structural geometric properties of MICP-R sets, which strongly differentiate them from the class of mixed-integer linear representable sets (MILP-R). First, we provide an example of an MICP-R set which is the countably infinite union of convex sets with countably infinitely many different recession cones. This is in sharp contrast with MILP-R sets which are at most infinite unions of polyhedra that share the same recession cone. Second, we provide an example of an MICP-R set which is the countably infinite union of polytopes all of which have different shapes (no pair is combinatorially equivalent, which implies they are not affine transformations of each other). Again, this is in sharp contrast with MILP-R sets which are at most infinite unions of polyhedra that are all translations of a finite subset of themselves. Interestingly, we show that a countably infinite union of convex sets sharing the same volume can be MICP-R only if the sets are all translations of a finite subset of themselves (i.e. the natural conceptual analogue to the MILP-R case).

We focus on the high-dimensional linear regression problem, where the algorithmic goal is to efficiently infer an unknown feature vector $\beta^*\in\mathbb{R}^p$ from its linear measurements, using a small number $n$ of samples. Unlike most of the literature, we make no sparsity assumption on $\beta^*$, but instead adopt a different regularization: In the noiseless setting, we assume $\beta^*$ consists of entries, which are either rational numbers with a common denominator $Q\in\mathbb{Z}^+$ (referred to as $Q$-rationality); or irrational numbers supported on a rationally independent set of bounded cardinality, known to learner; collectively called as the mixed-support assumption. Using a novel combination of the PSLQ integer relation detection, and LLL lattice basis reduction algorithms, we propose a polynomial-time algorithm which provably recovers a $\beta^*\in\mathbb{R}^p$ enjoying the mixed-support assumption, from its linear measurements $Y=X\beta^*\in\mathbb{R}^n$ for a large class of distributions for the random entries of $X$, even with one measurement $(n=1)$. In the noisy setting, we propose a polynomial-time, lattice-based algorithm, which recovers a $\beta^*\in\mathbb{R}^p$ enjoying $Q$-rationality, from its noisy measurements $Y=X\beta^*+W\in\mathbb{R}^n$, even with a single sample $(n=1)$. We further establish for large $Q$, and normal noise, this algorithm tolerates information-theoretically optimal level of noise. We then apply these ideas to develop a polynomial-time, single-sample algorithm for the phase retrieval problem. Our methods address the single-sample $(n=1)$ regime, where the sparsity-based methods such as LASSO and Basis Pursuit are known to fail. Furthermore, our results also reveal an algorithmic connection between the high-dimensional linear regression problem, and the integer relation detection, randomized subset-sum, and shortest vector problems., Comment: 56 pages. Parts of the material of this manuscript were presented at NeurIPS 2018, and ISIT 2019. This submission subsumes the content of arXiv:1803.06716

© 2019 IEEE. We consider the Gaussian multiple-access channel with two critical departures from the classical asymptotics: a) number of users proportional to block-length and b) each user sends a fixed number of data bits. We provide improved bounds on the tradeoff between the user density and the energy-per-bit. Interestingly, in this information-theoretic problem we rely on Gordon's lemma from Gaussian process theory. From the engineering standpoint, we discover a surprising new effect: good coded-access schemes can achieve perfect multi-user interference cancellation at low user density.In addition, by a similar method we analyze the limits of false-discovery in binary sparse regression problem in the asymptotic regime of number of measurements going to infinity at fixed ratios with problem dimension, sparsity and noise level. Our rigorous bound matches the formal replica-method prediction for some range of parameters with imperceptible numerical precision., NSF (Awards CCF-12-53205, CCF- 17-17842)

The deployment of massive MIMO systems has revived much of the interest in the study of the large-system performance of multiuser detection systems. In this paper, we prove a non-trivial upper bound on the bit-error rate (BER) of the MAP detector for BPSK signal transmission and equal-power condition. In particular, our bound is approximately tight at high-SNR. The proof is simple and relies on Gordon's comparison inequality. Interestingly, we show that under the assumption that Gordon's inequality is tight, the resulting BER prediction matches that of the replica method under the replica symmetry (RS) ansatz. Also, we prove that, when the ratio of receive to transmit antennas exceeds $0.9251$, the replica prediction matches the matched filter lower bound (MFB) at high-SNR. We corroborate our results by numerical evidence., Comment: To appear in ICASSP 2019

We consider the linear regression problem of estimating a $p$-dimensional vector $\beta$ from $n$ observations $Y = X \beta + W$, where $\beta_j \stackrel{\text{i.i.d.}}{\sim} \pi$ for a real-valued distribution $\pi$ with zero mean and unit variance, $X_{ij} \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,1)$, and $W_i\stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2)$. In the asymptotic regime where $n/p \to \delta$ and $ p/ \sigma^2 \to \mathsf{snr}$ for two fixed constants $\delta, \mathsf{snr}\in (0, \infty)$ as $p \to \infty$, the limiting (normalized) minimum mean-squared error (MMSE) has been characterized by the MMSE of an associated single-letter (additive Gaussian scalar) channel. In this paper, we show that if the MMSE function of the single-letter channel converges to a step function, then the limiting MMSE of estimating $\beta$ in the linear regression problem converges to a step function which jumps from $1$ to $0$ at a critical threshold. Moreover, we establish that the limiting mean-squared error of the (MSE-optimal) approximate message passing algorithm also converges to a step function with a larger threshold, providing evidence for the presence of a computational-statistical gap between the two thresholds., Comment: This paper was presented at the 2019 IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Guadeloupe, French West Indies

Motivated by growing concerns over ensuring privacy on social networks, we develop new algorithms and impossibility results for fitting complex statistical models to network data subject to rigorous privacy guarantees. We consider the so-called node-differentially private algorithms, which compute information about a graph or network while provably revealing almost no information about the presence or absence of a particular node in the graph. We provide new algorithms for node-differentially private estimation for a popular and expressive family of network models: stochastic block models and their generalization, graphons. Our algorithms improve on prior work, reducing their error quadratically and matching, in many regimes, the optimal nonprivate algorithm. We also show that for the simplest random graph models ($G(n,p)$ and $G(n,m)$), node-private algorithms can be qualitatively more accurate than for more complex models---converging at a rate of $\frac{1}{\epsilon^2 n^{3}}$ instead of $\frac{1}{\epsilon^2 n^2}$. This result uses a new extension lemma for differentially private algorithms that we hope will be broadly useful.

We present a sufficient condition to ensure the density of the set of rational functions with prescribed poles in the algebra A ∞ (Ω).

First we establish some generic universalities for Pade approximants in the closure X∞(Ω) in the space A∞(Ω) of all rational functions with poles off Ω¯. The closure Ω¯ of the domain Ω⊂C is taken with respect to the finite plane C. Next we give sufficient conditions on Ω so that X∞(Ω)=A∞(Ω). Some of these conditions imply that, even if the boundary ∂Ω of a Jordan domain Ω has infinite length, the integration operator on Ω preserves H∞(Ω) and A(Ω) as well. We also give an example of a Jordan domain Ω and a function f∈A(Ω), such that its antiderivative is not bounded on Ω. Finally we restate these results for Volterra operators on the open unit disc D and we complete them by some generic results.