Deflation conjecture and local dimensions of Brent equations

In this paper, a classical deflation process raised by Dayton, Li and Zeng is realized for the Brent equations, which provides new bounds for local dimensions of the solution set. Originally, this deflation process focuses on isolated solutions. We generalize it to the case of irreducible components and a related conjecture is given. We analysis its realization and apply it to the Brent equations. The decrease of the nullities is easily observed. So the deflation process can be served as a useful tool for determining the local dimensions. In addition, our result implies that along with the decrease of the tensor rank, the singular solutions will become more and more.Comment: Welcome to any comment

A Data-Driven Evolutionary Transfer Optimization for Expensive Problems in Dynamic Environments

Many real-world problems are usually computationally costly and the objective functions evolve over time. Data-driven, a.k.a. surrogate-assisted, evolutionary optimization has been recognized as an effective approach for tackling expensive black-box optimization problems in a static environment whereas it has rarely been studied under dynamic environments. This paper proposes a simple but effective transfer learning framework to empower data-driven evolutionary optimization to solve dynamic optimization problems. Specifically, it applies a hierarchical multi-output Gaussian process to capture the correlation between data collected from different time steps with a linearly increased number of hyperparameters. Furthermore, an adaptive source task selection along with a bespoke warm staring initialization mechanisms are proposed to better leverage the knowledge extracted from previous optimization exercises. By doing so, the data-driven evolutionary optimization can jump start the optimization in the new environment with a strictly limited computational budget. Experiments on synthetic benchmark test problems and a real-world case study demonstrate the effectiveness of our proposed algorithm against nine state-of-the-art peer algorithms

Block Edit Errors with Transpositions: Deterministic Document Exchange Protocols and Almost Optimal Binary Codes

Document exchange and error correcting codes are two fundamental problems regarding communications. In the first problem, Alice and Bob each holds a string, and the goal is for Alice to send a short sketch to Bob, so that Bob can recover Alice\u27s string. In the second problem, Alice sends a message with some redundant information to Bob through a channel that can add adversarial errors, and the goal is for Bob to correctly recover the message despite the errors. In both problems, an upper bound is placed on the number of errors between the two strings or that the channel can add, and a major goal is to minimize the size of the sketch or the redundant information. In this paper we focus on deterministic document exchange protocols and binary error correcting codes. Both problems have been studied extensively. In the case of Hamming errors (i.e., bit substitutions) and bit erasures, we have explicit constructions with asymptotically optimal parameters. However, other error types are still rather poorly understood. In a recent work [Kuan Cheng et al., 2018], the authors constructed explicit deterministic document exchange protocols and binary error correcting codes for edit errors with almost optimal parameters. Unfortunately, the constructions in [Kuan Cheng et al., 2018] do not work for other common errors such as block transpositions. In this paper, we generalize the constructions in [Kuan Cheng et al., 2018] to handle a much larger class of errors. These include bursts of insertions and deletions, as well as block transpositions. Specifically, we consider document exchange and error correcting codes where the total number of block insertions, block deletions, and block transpositions is at most k <= alpha n/log n for some constant 0<alpha<1. In addition, the total number of bits inserted and deleted by the first two kinds of operations is at most t <= beta n for some constant 0<beta<1, where n is the length of Alice\u27s string or message. We construct explicit, deterministic document exchange protocols with sketch size O((k log n +t) log^2 n/{k log n + t}) and explicit binary error correcting code with O(k log n log log log n+t) redundant bits. As a comparison, the information-theoretic optimum for both problems is Theta(k log n+t). As far as we know, previously there are no known explicit deterministic document exchange protocols in this case, and the best known binary code needs Omega(n) redundant bits even to correct just one block transposition [L. J. Schulman and D. Zuckerman, 1999]