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We introduce and analyse an efficient decoder for the quantum Tanner codes of that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight $O(\sqrt{n \log n})$. We also work on the link between quantum Tanner codes and the Lifted Product codes of Panteleev and Kalachev, and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.

Designing neural networks with bounded Lipschitz constant is a promising way to obtain certifiably robust classifiers against adversarial examples. However, the relevant progress for the important $\ell_\infty$ perturbation setting is rather limited, and a principled understanding of how to design expressive $\ell_\infty$ Lipschitz networks is still lacking. In this paper, we bridge the gap by studying certified $\ell_\infty$ robustness from a novel perspective of representing Boolean functions. We derive two fundamental impossibility results that hold for any standard Lipschitz network: one for robust classification on finite datasets, and the other for Lipschitz function approximation. These results identify that networks built upon norm-bounded affine layers and Lipschitz activations intrinsically lose expressive power even in the two-dimensional case, and shed light on how recently proposed Lipschitz networks (e.g., GroupSort and $\ell_\infty$-distance nets) bypass these impossibilities by leveraging order statistic functions. Finally, based on these insights, we develop a unified Lipschitz network that generalizes prior works, and design a practical version that can be efficiently trained (making certified robust training free). Extensive experiments show that our approach is scalable, efficient, and consistently yields better certified robustness across multiple datasets and perturbation radii than prior Lipschitz networks. Our code is available at //github.com/zbh2047/SortNet.

The problem of fair division of indivisible goods has been receiving much attention recently. The prominent metric of envy-freeness can always be satisfied in the divisible goods setting (see for example \cite{BT95}), but often cannot be satisfied in the indivisible goods setting. This has led to many relaxations thereof being introduced. We study the existence of {\em maximin share (MMS)} allocations, which is one such relaxation. Previous work has shown that MMS allocations are guaranteed to exist for all instances with $n$ players and $m$ goods if $m \leq n+4$. We extend this guarantee to the case of $m = n+5$ and show that the same guarantee fails for $m = n+6$.

This paper is concerned with low-rank matrix optimization, which has found a wide range of applications in machine learning. This problem in the special case of matrix sensing has been studied extensively through the notion of Restricted Isometry Property (RIP), leading to a wealth of results on the geometric landscape of the problem and the convergence rate of common algorithms. However, the existing results can handle the problem in the case with a general objective function subject to noisy data only when the RIP constant is close to 0. In this paper, we develop a new mathematical framework to solve the above-mentioned problem with a far less restrictive RIP constant. We prove that as long as the RIP constant of the noiseless objective is less than $1/3$, any spurious local solution of the noisy optimization problem must be close to the ground truth solution. By working through the strict saddle property, we also show that an approximate solution can be found in polynomial time. We characterize the geometry of the spurious local minima of the problem in a local region around the ground truth in the case when the RIP constant is greater than $1/3$. Compared to the existing results in the literature, this paper offers the strongest RIP bound and provides a complete theoretical analysis on the global and local optimization landscapes of general low-rank optimization problems under random corruptions from any finite-variance family.

We consider the problem of finding the matching map between two sets of $d$-dimensional noisy feature-vectors. The distinctive feature of our setting is that we do not assume that all the vectors of the first set have their corresponding vector in the second set. If $n$ and $m$ are the sizes of these two sets, we assume that the matching map that should be recovered is defined on a subset of unknown cardinality $k^*\le \min(n,m)$. We show that, in the high-dimensional setting, if the signal-to-noise ratio is larger than $5(d\log(4nm/\alpha))^{1/4}$, then the true matching map can be recovered with probability $1-\alpha$. Interestingly, this threshold does not depend on $k^*$ and is the same as the one obtained in prior work in the case of $k = \min(n,m)$. The procedure for which the aforementioned property is proved is obtained by a data-driven selection among candidate mappings $\{\hat\pi_k:k\in[\min(n,m)]\}$. Each $\hat\pi_k$ minimizes the sum of squares of distances between two sets of size $k$. The resulting optimization problem can be formulated as a minimum-cost flow problem, and thus solved efficiently. Finally, we report the results of numerical experiments on both synthetic and real-world data that illustrate our theoretical results and provide further insight into the properties of the algorithms studied in this work.

The goal of Bayesian deep learning is to provide uncertainty quantification via the posterior distribution. However, exact inference over the weight space is computationally intractable due to the ultra-high dimensions of the neural network. Variational inference (VI) is a promising approach, but naive application on weight space does not scale well and often underperform on predictive accuracy. In this paper, we propose a new adaptive variational Bayesian algorithm to train neural networks on weight space that achieves high predictive accuracy. By showing that there is an equivalence to Stochastic Gradient Hamiltonian Monte Carlo(SGHMC) with preconditioning matrix, we then propose an MCMC within EM algorithm, which incorporates the spike-and-slab prior to capture the sparsity of the neural network. The EM-MCMC algorithm allows us to perform optimization and model pruning within one-shot. We evaluate our methods on CIFAR-10, CIFAR-100 and ImageNet datasets, and demonstrate that our dense model can reach the state-of-the-art performance and our sparse model perform very well compared to previously proposed pruning schemes.

Quantum cloud computing is a promising paradigm for efficiently provisioning quantum resources (i.e., qubits) to users. In quantum cloud computing, quantum cloud providers provision quantum resources in reservation and on-demand plans for users. Literally, the cost of quantum resources in the reservation plan is expected to be cheaper than the cost of quantum resources in the on-demand plan. However, quantum resources in the reservation plan have to be reserved in advance without information about the requirement of quantum circuits beforehand, and consequently, the resources are insufficient, i.e., under-reservation. Hence, quantum resources in the on-demand plan can be used to compensate for the unsatisfied quantum resources required. To end this, we propose a quantum resource allocation for the quantum cloud computing system in which quantum resources and the minimum waiting time of quantum circuits are jointly optimized. Particularly, the objective is to minimize the total costs of quantum circuits under uncertainties regarding qubit requirement and minimum waiting time of quantum circuits. In experiments, practical circuits of quantum Fourier transform are applied to evaluate the proposed qubit resource allocation. The results illustrate that the proposed qubit resource allocation can achieve the optimal total costs.

In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.

This paper presents a succinct review of attempts in the literature to use game theory to model decision making scenarios relevant to defence applications. Game theory has been proven as a very effective tool in modelling decision making processes of intelligent agents, entities, and players. It has been used to model scenarios from diverse fields such as economics, evolutionary biology, and computer science. In defence applications, there is often a need to model and predict actions of hostile actors, and players who try to evade or out-smart each other. Modelling how the actions of competitive players shape the decision making of each other is the forte of game theory. In past decades, there have been several studies which applied different branches of game theory to model a range of defence-related scenarios. This paper provides a structured review of such attempts, and classifies existing literature in terms of the kind of warfare modelled, the types of game used, and the players involved. The presented analysis provides a concise summary about the state-of-the-art with regards to the use of game theory in defence applications, and highlights the benefits and limitations of game theory in the considered scenarios.

Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.