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Correlation Clustering is an important clustering problem with many applications. We study the reconstruction version of this problem in which one is seeking to reconstruct a latent clustering that has been corrupted by random noise and adversarial modifications. Concerning the latter, there is a standard "post-adversarial" model in the literature, in which adversarial modifications come after the noise. Here, we introduce and analyse a "pre-adversarial" model in which adversarial modifications come before the noise. Given an input coming from such a semi-adversarial generative model, the goal is to reconstruct almost perfectly and with high probability the latent clustering. We focus on the case where the hidden clusters have nearly equal size and show the following. In the pre-adversarial setting, spectral algorithms are optimal, in the sense that they reconstruct all the way to the information-theoretic threshold beyond which no reconstruction is possible. This is in contrast to the post-adversarial setting, in which their ability to restore the hidden clusters stops before the threshold, but the gap is optimally filled by SDP-based algorithms. These results highlight a heretofore unknown robustness of spectral algorithms, showing them less brittle than previously thought.

Let $f$ be a polynomial of degree $d$ in $n$ variables over a finite field $\mathbb{F}$. The polynomial is said to be unbiased if the distribution of $f(x)$ for a uniform input $x \in \mathbb{F}^n$ is close to the uniform distribution over $\mathbb{F}$, and is called biased otherwise. The polynomial is said to have low rank if it can be expressed as a composition of a few lower degree polynomials. Green and Tao [Contrib. Discrete Math 2009] and Kaufman and Lovett [FOCS 2008] showed that bias implies low rank for fixed degree polynomials over fixed prime fields. This lies at the heart of many tools in higher order Fourier analysis. In this work, we extend this result to all prime fields (of size possibly growing with $n$). We also provide a generalization to nonprime fields in the large characteristic case. However, we state all our applications in the prime field setting for the sake of simplicity of presentation. Using the above generalization to large fields as a starting point, we are also able to settle the list decoding radius of fixed degree Reed-Muller codes over growing fields. The case of fixed size fields was solved by Bhowmick and Lovett [STOC 2015], which resolved a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008]. Here, we show that the list decoding radius is equal the minimum distance of the code for all fixed degrees, even when the field size is possibly growing with $n$. Additionally, we effectively resolve the weight distribution problem for Reed-Muller codes of fixed degree over all fields, first raised in 1977 in the classic textbook by MacWilliams and Sloane [Research Problem 15.1 in Theory of Error Correcting Codes].

Local electricity markets represent a way of supplementing traditional retailing contracts for end consumers -- among these markets, the renewable energy community has gained momentum over the last few years. This paper proposes a practical and readily to be adopted modelling solution for these communities, one that allows their members to share the economic benefits derived from them. The proposed solution relies on an \emph{ex-post} allocation of the electricity that is generated within energy communities (i.e., local electricity) based on the optimisation of \emph{repartition keys}. Repartition keys are therefore optimally computed to represent the proportion of total local electricity to be allocated to each community member, and aim to minimise the sum of electricity bills of all community members. Since the optimisation takes place \emph{ex-post} the repartition keys do not modify the actual electricity flows, but rather the financial flows of the community members. Then, the billing process of the community will take these keys into account to correctly send the electricity bills to each member. Building on this concept, we also introduce two additions to the basic algorithm to enhance the stability of the community, which a global bill minimisation may fail to ensure (e.g., very asymmetrical solutions between members may lead to some of them opting out).

Shared spectrum systems facilitate spectrum allocation to unlicensed users without harming the licensed users; they offer great promise in optimizing spectrum utility, but their management (in particular, efficient spectrum allocation to unlicensed users) is challenging. A significant shortcoming of current allocation methods is that they are either done very conservatively to ensure correctness, or are based on imperfect propagation models and/or spectrum sensing with poor spatial granularity. This leads to poor spectrum utilization, the fundamental objective of shared spectrum systems. To allocate spectrum near-optimally to secondary users in general scenarios, we fundamentally need to have knowledge of the signal path-loss function. In practice, however, even the best known path-loss models have unsatisfactory accuracy, and conducting extensive surveys to gather path-loss values is infeasible. To circumvent this challenge, we propose to learn the spectrum allocation function directly using supervised learning techniques. We particularly address the scenarios when the primary users' information may not be available; for such settings, we make use of a crowdsourced sensing architecture and use the spectrum sensor readings as features. We develop an efficient CNN-based approach (called DeepAlloc) and address various challenges that arise in its application to the learning the spectrum allocation function. Via extensive large-scale simulation and a small testbed, we demonstrate the effectiveness of our developed techniques; in particular, we observe that our approach improves the accuracy of standard learning techniques and prior work by up to 60%.

We propose in this work to employ the Box-LASSO, a variation of the popular LASSO method, as a low-complexity decoder in a massive multiple-input multiple-output (MIMO) wireless communication system. The Box-LASSO is mainly useful for detecting simultaneously structured signals such as signals that are known to be sparse and bounded. One modulation technique that generates essentially sparse and bounded constellation points is the so-called generalized space-shift keying (GSSK) modulation. In this direction, we derive high dimensional sharp characterizations of various performance measures of the Box-LASSO such as the mean square error, probability of support recovery, and the element error rate, under independent and identically distributed (i.i.d.) Gaussian channels that are not perfectly known. In particular, the analytical characterizations can be used to demonstrate performance improvements of the Box-LASSO as compared to the widely used standard LASSO. Then, we can use these measures to optimally tune the involved hyper-parameters of Box-LASSO such as the regularization parameter. In addition, we derive optimum power allocation and training duration schemes in a training-based massive MIMO system. Monte Carlo simulations are used to validate these premises and to show the sharpness of the derived analytical results.

Image reconstruction based on indirect, noisy, or incomplete data remains an important yet challenging task. While methods such as compressive sensing have demonstrated high-resolution image recovery in various settings, there remain issues of robustness due to parameter tuning. Moreover, since the recovery is limited to a point estimate, it is impossible to quantify the uncertainty, which is often desirable. Due to these inherent limitations, a sparse Bayesian learning approach is sometimes adopted to recover a posterior distribution of the unknown. Sparse Bayesian learning assumes that some linear transformation of the unknown is sparse. However, most of the methods developed are tailored to specific problems, with particular forward models and priors. Here, we present a generalized approach to sparse Bayesian learning. It has the advantage that it can be used for various types of data acquisitions and prior information. Some preliminary results on image reconstruction/recovery indicate its potential use for denoising, deblurring, and magnetic resonance imaging.

We consider performance enhancement of asymmetrically-clipped optical orthogonal frequency division multiplexing (ACO-OFDM) and related optical OFDM schemes, which are variations of OFDM in intensity-modulated optical wireless communications. Unlike most existing studies on specific designs of improved receivers, this paper investigates information theoretic limits of all possible receivers. For independent and identically distributed complex Gaussian inputs, we obtain an exact characterization of information rate of ACO-OFDM with improved receivers for all SNRs. It is proved that the high-SNR gain of improved receivers asymptotically achieve 1/4 bits per channel use, which is equivalent to 3 dB in electrical SNR or 1.5 dB in optical SNR; as the SNR decreases, the maximum achievable SNR gain of improved receivers decreases monotonically to a non-zero low-SNR limit, corresponding to an information rate gain of 36.3%. For practically used constellations, we derive an upper bound on the gain of improved receivers. Numerical results demonstrate that the upper bound can be approached to within 1 dB in optical SNR by combining existing improved receivers and coded modulation. We also show that our information theoretic analyses can be extended to Flip-OFDM and PAM-DMT. Our results imply that, for the considered schemes, improved receivers may reduce the gap to channel capacity significantly at low-to-moderate SNR.

The impact of device and circuit-level effects in mixed-signal Resistive Random Access Memory (RRAM) accelerators typically manifest as performance degradation of Deep Learning (DL) algorithms, but the degree of impact varies based on algorithmic features. These include network architecture, capacity, weight distribution, and the type of inter-layer connections. Techniques are continuously emerging to efficiently train sparse neural networks, which may have activation sparsity, quantization, and memristive noise. In this paper, we present an extended Design Space Exploration (DSE) methodology to quantify the benefits and limitations of dense and sparse mapping schemes for a variety of network architectures. While sparsity of connectivity promotes less power consumption and is often optimized for extracting localized features, its performance on tiled RRAM arrays may be more susceptible to noise due to under-parameterization, when compared to dense mapping schemes. Moreover, we present a case study quantifying and formalizing the trade-offs of typical non-idealities introduced into 1-Transistor-1-Resistor (1T1R) tiled memristive architectures and the size of modular crossbar tiles using the CIFAR-10 dataset.

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.

The Transformer is widely used in natural language processing tasks. To train a Transformer however, one usually needs a carefully designed learning rate warm-up stage, which is shown to be crucial to the final performance but will slow down the optimization and bring more hyper-parameter tunings. In this paper, we first study theoretically why the learning rate warm-up stage is essential and show that the location of layer normalization matters. Specifically, we prove with mean field theory that at initialization, for the original-designed Post-LN Transformer, which places the layer normalization between the residual blocks, the expected gradients of the parameters near the output layer are large. Therefore, using a large learning rate on those gradients makes the training unstable. The warm-up stage is practically helpful for avoiding this problem. On the other hand, our theory also shows that if the layer normalization is put inside the residual blocks (recently proposed as Pre-LN Transformer), the gradients are well-behaved at initialization. This motivates us to remove the warm-up stage for the training of Pre-LN Transformers. We show in our experiments that Pre-LN Transformers without the warm-up stage can reach comparable results with baselines while requiring significantly less training time and hyper-parameter tuning on a wide range of applications.