Non-maximum suppression (NMS) is an indispensable post-processing step in object detection. With the continuous optimization of network models, NMS has become the "last mile" to enhance the efficiency of object detection.

Many areas of machine learning and science involve large linear algebra problems, such as eigendecompositions, solving linear systems, computing matrix exponentials, and trace estimation. The matrices involved often have Kronecker, convolutional, block diagonal, sum, or product structure. In this paper, we propose a simple but general framework for large-scale linear algebra problems in machine learning, named CoLA (Compositional Linear Algebra).

Since the weak convergence for stochastic processes does not account for the growth of information over time which is represented by the underlying filtration, a slightly erroneous stochastic model in weak topology may cause huge loss in multi-periods decision making problems. To address such discontinuities Aldous introduced the extended weak convergence, which can fully characterise all essential properties, including the filtration, of stochastic processes; however was considered to be hard to find efficient numerical implementations. In this paper, we introduce a novel metric called High Rank PCF Distance (HRPCFD) for extended weak convergence based on the high rank path development method from rough path theory, which also defines the characteristic function for measure-valued processes. We then show that such HRPCFD admits many favourable analytic properties which allows us to design an efficient algorithm for training HRPCFD from data and construct the HRPCF-GAN by using HRPCFD as the discriminator for conditional time series generation. Our numerical experiments on both hypothesis testing and generative modelling validate the out-performance of our approach compared with several state-of-the-art methods, highlighting its potential in broad applications of synthetic time series generation and in addressing classic financial and economic challenges, such as optimal stopping or utility maximisation problems.

We propose and analyse a novel statistical procedure, coined AgraSSt, to assess the quality of graph generators which may not be available in explicit forms. In particular, AgraSSt can be used to determine whether a learned graph generating process is capable of generating graphs which resemble a given input graph. Inspired by Stein operators for random graphs, the key idea of AgraSSt is the construction of a kernel discrepancy based on an operator obtained from the graph generator. AgraSSt can provide interpretable criticisms for a graph generator training procedure and help identify reliable sample batches for downstream tasks. We give theoretical guarantees for a broad class of random graph models. Moreover, we provide empirical results on both synthetic input graphs with known graph generation procedures, and real-world input graphs that the state-of-the-art (deep) generative models for graphs are trained on.

Stochastic Programming is a powerful modeling framework for decision-making under uncertainty. In this work, we tackle two-stage stochastic programs (2SPs), the most widely used class of stochastic programming models. Solving 2SPs exactly requires optimizing over an expected value function that is computationally intractable. Having a mixed-integer linear program (MIP) or a nonlinear program (NLP) in the second stage further aggravates the intractability, even when specialized algorithms that exploit problem structure are employed. Finding high-quality (first-stage) solutions - without leveraging problem structure - can be crucial in such settings.

For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than 1/e require continuous ideas, such as queries to the multilinear extension of a submodular function and its gradient, which are typically expensive to simulate with the original set function. For combinatorial algorithms, the best known approximation ratios for both size and matroid constraint are obtained by a simple randomized greedy algorithm of Buchbinder et al. [9]: 1/e 0.367 for size constraint and 0.281 for the matroid constraint in O(kn) queries, where k is the rank of the matroid. In this work, we develop the first combinatorial algorithms to break the 1/e barrier: we obtain approximation ratio of 0.385 in O(kn) queries to the submodular set function for size constraint, and 0.305 for a general matroid constraint. These are achieved by guiding the randomized greedy algorithm with a fast local search algorithm. Further, we develop deterministic versions of these algorithms, maintaining the same ratio and asymptotic time complexity. Finally, we develop a deterministic, nearly linear time algorithm with ratio 0.377.

Distributed sparse learning with a cluster of multiple machines has attracted much attention in machine learning, especially for large-scale applications with high-dimensional data.

In this paper, we study private sparsification of graphs. In particular, we give an algorithm that given an input graph, returns a sparse graph which approximates the spectrum of the input graph while ensuring differential privacy. This allows one to solve many graph problems privately yet efficiently and accurately. This is exemplified with application of the proposed meta-algorithm to graph algorithms for privately answering cut-queries, as well as practical algorithms for computing MAX-CUT and SPARSEST-CUT with better accuracy than previously known. We also give an efficient private algorithm to learn Laplacian eigenmap on a graph.

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