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$n$-cycle permutations with small $n$ have the advantage that their compositional inverses are efficient in terms of implementation. They can be also used in constructing Bent functions and designing codes. Since the AGW Criterion was proposed, the permuting property of several forms of polynomials has been studied. In this paper, characterizations of several types of $n$-cycle permutations are investigated. Three criteria for $ n $-cycle permutations of the form $xh(\lambda(x))$, $ h(\psi(x)) \varphi(x)+g(\psi(x)) $ and $g\left( x^{q^i} -x +\delta \right) +bx $ with general $n$ are provided. We demonstrate these criteria by providing explicit constructions. For the form of $x^rh(x^s)$, several new explicit triple-cycle permutations are also provided. Finally, we also consider triple-cycle permutations of the form $x^t + c\rm Tr_{q^m/q}(x^s)$ and provide one explicit construction. Many of our constructions are both new in the $n$-cycle property and the permutation property.

Multiple Tensor-Times-Matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. We establish communication lower bounds that determine how much data movement is required to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor-times-matrix operations.

Let $\mathbf{H}$ be the Cartesian product of a family of finite abelian groups. Via a polynomial approach, we give sufficient conditions for a partition of $\mathbf{H}$ induced by weighted poset metric to be reflexive, which also become necessary for some special cases. Moreover, by examining the roots of the Krawtchouk polynomials, we establish non-reflexive partitions of $\mathbf{H}$ induced by combinatorial metric. When $\mathbf{H}$ is a vector space over a finite field $\mathbb{F}$, we consider the property of admitting MacWilliams identity (PAMI) and the MacWilliams extension property (MEP) for partitions of $\mathbf{H}$. With some invariance assumptions, we show that two partitions of $\mathbf{H}$ admit MacWilliams identity if and only if they are mutually dual and reflexive, and any partition of $\mathbf{H}$ satisfying the MEP is in fact an orbit partition induced by some subgroup of $\Aut_{\mathbb{F}}(\mathbf{H})$, which is necessarily reflexive. As an application of the aforementioned results, we establish partitions of $\mathbf{H}$ induced by combinatorial metric that do not satisfy the MEP, which further enable us to provide counter-examples to a conjecture proposed by Pinheiro, Machado and Firer in \cite{39}.

We consider the problem of estimating the factors of a rank-$1$ matrix with i.i.d. Gaussian, rank-$1$ measurements that are nonlinearly transformed and corrupted by noise. Considering two prototypical choices for the nonlinearity, we study the convergence properties of a natural alternating update rule for this nonconvex optimization problem starting from a random initialization. We show sharp convergence guarantees for a sample-split version of the algorithm by deriving a deterministic recursion that is accurate even in high-dimensional problems. Notably, while the infinite-sample population update is uninformative and suggests exact recovery in a single step, the algorithm -- and our deterministic prediction -- converges geometrically fast from a random initialization. Our sharp, non-asymptotic analysis also exposes several other fine-grained properties of this problem, including how the nonlinearity and noise level affect convergence behavior. On a technical level, our results are enabled by showing that the empirical error recursion can be predicted by our deterministic sequence within fluctuations of the order $n^{-1/2}$ when each iteration is run with $n$ observations. Our technique leverages leave-one-out tools originating in the literature on high-dimensional $M$-estimation and provides an avenue for sharply analyzing higher-order iterative algorithms from a random initialization in other high-dimensional optimization problems with random data.

Federated learning~(FL) has recently attracted increasing attention from academia and industry, with the ultimate goal of achieving collaborative training under privacy and communication constraints. Existing iterative model averaging based FL algorithms require a large number of communication rounds to obtain a well-performed model due to extremely unbalanced and non-i.i.d data partitioning among different clients. Thus, we propose FedDM to build the global training objective from multiple local surrogate functions, which enables the server to gain a more global view of the loss landscape. In detail, we construct synthetic sets of data on each client to locally match the loss landscape from original data through distribution matching. FedDM reduces communication rounds and improves model quality by transmitting more informative and smaller synthesized data compared with unwieldy model weights. We conduct extensive experiments on three image classification datasets, and results show that our method can outperform other FL counterparts in terms of efficiency and model performance. Moreover, we demonstrate that FedDM can be adapted to preserve differential privacy with Gaussian mechanism and train a better model under the same privacy budget.

Federated learning (FL) is a recently developed area of machine learning, in which the private data of a large number of distributed clients is used to develop a global model under the coordination of a central server without explicitly exposing the data. The standard FL strategy has a number of significant bottlenecks including large communication requirements and high impact on the clients' resources. Several strategies have been described in the literature trying to address these issues. In this paper, a novel scheme based on the notion of "model growing" is proposed. Initially, the server deploys a small model of low complexity, which is trained to capture the data complexity during the initial set of rounds. When the performance of such a model saturates, the server switches to a larger model with the help of function-preserving transformations. The model complexity increases as more data is processed by the clients, and the overall process continues until the desired performance is achieved. Therefore, the most complex model is broadcast only at the final stage in our approach resulting in substantial reduction in communication cost and client computational requirements. The proposed approach is tested extensively on three standard benchmarks and is shown to achieve substantial reduction in communication and client computation while achieving comparable accuracy when compared to the current most effective strategies.

Given two relations containing multiple measurements - possibly with uncertainties - our objective is to find which sets of attributes from the first have a corresponding set on the second, using exclusively a sample of the data. This approach could be used even when the associated metadata is damaged, missing or incomplete, or when the volume is too big for exact methods. This problem is similar to the search of Inclusion Dependencies (IND), a type of rule over two relations asserting that for a set of attributes X from the first, every combination of values appears on a set Y from the second. Existing IND can be found exploiting the existence of a partial order relation called specialization. However, this relation is based on set theory, requiring the values to be directly comparable. Statistical tests are an intuitive possible replacement, but it has not been studied how would they affect the underlying assumptions. In this paper we formally review the effect that a statistical approach has over the inference rules applied to IND discovery. Our results confirm the intuitive thought that statistical tests can be used, but not in a directly equivalent manner. We provide a workable alternative based on a "hierarchy of null hypotheses", allowing for the automatic discovery of multi-dimensional equally distributed sets of attributes.

Private set intersection (PSI) allows two mutually untrusting parties to compute an intersection of their sets, without revealing information about items that are not in the intersection. This work introduces a PSI variant called distance-aware PSI (DA-PSI) for sets whose elements lie in a metric space. DA-PSI returns pairs of items that are within a specified distance threshold of each other. This paper puts forward DA-PSI constructions for two metric spaces: (i) Minkowski distance of order 1 over the set of integers (i.e., for integers $a$ and $b$, their distance is $|a-b|$); and (ii) Hamming distance over the set of binary strings of length $\ell$. In the Minkowski DA-PSI protocol, the communication complexity scales logarithmically in the distance threshold and linearly in the set size. In the Hamming DA-PSI protocol, the communication volume scales quadratically in the distance threshold and is independent of the dimensionality of string length $\ell$. Experimental results with real applications confirm that DA-PSI provides more effective matching at lower cost than naive solutions.

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.

Recommender systems (RSs) have been the most important technology for increasing the business in Taobao, the largest online consumer-to-consumer (C2C) platform in China. The billion-scale data in Taobao creates three major challenges to Taobao's RS: scalability, sparsity and cold start. In this paper, we present our technical solutions to address these three challenges. The methods are based on the graph embedding framework. We first construct an item graph from users' behavior history. Each item is then represented as a vector using graph embedding. The item embeddings are employed to compute pairwise similarities between all items, which are then used in the recommendation process. To alleviate the sparsity and cold start problems, side information is incorporated into the embedding framework. We propose two aggregation methods to integrate the embeddings of items and the corresponding side information. Experimental results from offline experiments show that methods incorporating side information are superior to those that do not. Further, we describe the platform upon which the embedding methods are deployed and the workflow to process the billion-scale data in Taobao. Using online A/B test, we show that the online Click-Through-Rate (CTRs) are improved comparing to the previous recommendation methods widely used in Taobao, further demonstrating the effectiveness and feasibility of our proposed methods in Taobao's live production environment.