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Frameproof codes have been extensively studied for many years due to their application in copyright protection and their connection to extremal set theory. In this paper, we investigate upper bounds on the cardinality of wide-sense $t$-frameproof codes. For $t=2$, we apply results from Sperner theory to give a better upper bound, which significantly improves a recent bound by Zhou and Zhou. For $t\geq 3$, we provide a general upper bound by establishing a relation between wide-sense frameproof codes and cover-free families. Finally, when the code length $n$ is at most $\frac{15+\sqrt{33}}{24}(t-1)^2$, we show that a wide-sense $t$-frameproof code has at most $n$ codewords, and the unique optimal code consists of all weight-one codewords. As byproducts, our results improve several best known results on binary $t$-frameproof codes.

Bayesian networks are widely utilised in various fields, offering elegant representations of factorisations and causal relationships. We use surjective functions to reduce the dimensionality of the Bayesian networks by combining states and study the preservation of their factorisation structure. We introduce and define corresponding notions, analyse their properties, and provide examples of highly symmetric special cases, enhancing the understanding of the fundamental properties of such reductions for Bayesian networks. We also discuss the connection between this and reductions of homogeneous and non-homogeneous Markov chains.

Understanding the mechanisms through which neural networks extract statistics from input-label pairs is one of the most important unsolved problems in supervised learning. Prior works have identified that the gram matrices of the weights in trained neural networks of general architectures are proportional to the average gradient outer product of the model, in a statement known as the Neural Feature Ansatz (NFA). However, the reason these quantities become correlated during training is poorly understood. In this work, we explain the emergence of this correlation. We identify that the NFA is equivalent to alignment between the left singular structure of the weight matrices and a significant component of the empirical neural tangent kernels associated with those weights. We establish that the NFA introduced in prior works is driven by a centered NFA that isolates this alignment. We show that the speed of NFA development can be predicted analytically at early training times in terms of simple statistics of the inputs and labels. Finally, we introduce a simple intervention to increase NFA correlation at any given layer, which dramatically improves the quality of features learned.

Decoding of Low-Density Parity Check (LDPC) codes can be viewed as a special case of XOR-SAT problems, for which low-computational complexity bit-flipping algorithms have been proposed in the literature. However, a performance gap exists between the bit-flipping LDPC decoding algorithms and the benchmark LDPC decoding algorithms, such as the Sum-Product Algorithm (SPA). In this paper, we propose an XOR-SAT solver using log-sum-exponential functions and demonstrate its advantages for LDPC decoding. This is then approximated using the Margin Propagation formulation to attain a low-complexity LDPC decoder. The proposed algorithm uses soft information to decide the bit-flips that maximize the number of parity check constraints satisfied over an optimization function. The proposed solver can achieve results that are within $0.1$dB of the Sum-Product Algorithm for the same number of code iterations. It is also at least 10x lesser than other Gradient-Descent Bit Flipping decoding algorithms, which are also bit-flipping algorithms based on optimization functions. The approximation using the Margin Propagation formulation does not require any multipliers, resulting in significantly lower computational complexity than other soft-decision Bit-Flipping LDPC decoders.

We present a new decoder for the surface code, which combines the accuracy of the tensor-network decoders with the efficiency and parallelism of the belief-propagation algorithm. Our main idea is to replace the expensive tensor-network contraction step in the tensor-network decoders with the blockBP algorithm - a recent approximate contraction algorithm, based on belief propagation. Our decoder is therefore a belief-propagation decoder that works in the degenerate maximal likelihood decoding framework. Unlike conventional tensor-network decoders, our algorithm can run efficiently in parallel, and may therefore be suitable for real-time decoding. We numerically test our decoder and show that for a large range of lattice sizes and noise levels it delivers a logical error probability that outperforms the Minimal-Weight-Perfect-Matching (MWPM) decoder, sometimes by more than an order of magnitude.

We study a robust multi-agent multi-armed bandit problem where multiple clients or participants are distributed on a fully decentralized blockchain, with the possibility of some being malicious. The rewards of arms are homogeneous among the clients, following time-invariant stochastic distributions that are revealed to the participants only when the system is secure enough. The system's objective is to efficiently ensure the cumulative rewards gained by the honest participants. To this end and to the best of our knowledge, we are the first to incorporate advanced techniques from blockchains, as well as novel mechanisms, into the system to design optimal strategies for honest participants. This allows various malicious behaviors and the maintenance of participant privacy. More specifically, we randomly select a pool of validators who have access to all participants, design a brand-new consensus mechanism based on digital signatures for these validators, invent a UCB-based strategy that requires less information from participants through secure multi-party computation, and design the chain-participant interaction and an incentive mechanism to encourage participants' participation. Notably, we are the first to prove the theoretical guarantee of the proposed algorithms by regret analyses in the context of optimality in blockchains. Unlike existing work that integrates blockchains with learning problems such as federated learning which mainly focuses on numerical optimality, we demonstrate that the regret of honest participants is upper bounded by $log{T}$. This is consistent with the multi-agent multi-armed bandit problem without malicious participants and the robust multi-agent multi-armed bandit problem with purely Byzantine attacks.

In this note, we apply some techniques developed in [1]-[3] to give a particular construction of bivariate Abelian Codes from cyclic codes, multiplying their dimension and preserving their apparent distance. We show that, in the case of cyclic codes whose maximum BCH bound equals its minimum distance the obtained abelian code verifies the same property; that is, the strong apparent distance and the minimum distance coincide. We finally use this construction to multiply Reed-Solomon codes to abelian codes

By abstracting over well-known properties of De Bruijn's representation with nameless dummies, we design a new theory of syntax with variable binding and capture-avoiding substitution. We propose it as a simpler alternative to Fiore, Plotkin, and Turi's approach, with which we establish a strong formal link. We also show that our theory easily incorporates simple types and equations between terms.

Traditional stabilizer codes operate over prime power local-dimensions. In this work we extend the stabilizer formalism using the local-dimension-invariant setting to import stabilizer codes from these standard local-dimensions to other cases. In particular, we show that any traditional stabilizer code can be used for analog continuous-variable codes, and consider restrictions in phase space and discretized phase space. This puts this framework on an equivalent footing as traditional stabilizer codes. Following this, using extensions of prior ideas, we show that a stabilizer code originally designed with a finite field local-dimension can be transformed into a code with the same $n$, $k$, and $d$ parameters for any integral domain. This is of theoretical interest and can be of use for systems whose local-dimension is better described by mathematical rings, which permits the use of traditional stabilizer codes for protecting their information as well.

The recent proliferation of knowledge graphs (KGs) coupled with incomplete or partial information, in the form of missing relations (links) between entities, has fueled a lot of research on knowledge base completion (also known as relation prediction). Several recent works suggest that convolutional neural network (CNN) based models generate richer and more expressive feature embeddings and hence also perform well on relation prediction. However, we observe that these KG embeddings treat triples independently and thus fail to cover the complex and hidden information that is inherently implicit in the local neighborhood surrounding a triple. To this effect, our paper proposes a novel attention based feature embedding that captures both entity and relation features in any given entity's neighborhood. Additionally, we also encapsulate relation clusters and multihop relations in our model. Our empirical study offers insights into the efficacy of our attention based model and we show marked performance gains in comparison to state of the art methods on all datasets.