arXiv subjects
Publications and source records attributed to Chai Wah Wu.
We consider the minimal number of points on a regular grid on the plane that generates $n$ line segments of points of exactly length $k$. We illustrate how this is related to the $n$-queens problem on the toroidal chessboard and show that this number is upper bounded by $kn/3$ and approaches $kn/4$ as $n\rightarrow\infty$ when $k+1$ is coprime with $6$ or when $k$ is large.
The classical rearrangement inequality provides bounds for the sum of products of two sequences under permutations of terms and show that similarly ordered sequences provide the largest value whereas opposite ordered sequences provide the smallest value. This has been generalized to multiple sequences to show that similarly ordered sequences provide the largest value. However, the permutations of the sequences that result in the smallest value are generally not known. We show a variant of the rearrangement inequality for which a lower bound can be obtained and conditions for which this bound is achieved for a sequence of permutations. We also study a generalization of the rearrangement inequality and a variation where the permutations of terms can be across the various sequences. For this variation, we can also find the minimizing and maximizing sequences under certain conditions. Finally, we also look at rearrangement inequalities of other objects that can be ordered such as functions and matrices.
Hamiltonian learning is an important procedure in quantum system identification, calibration, and successful operation of quantum computers. Through queries to the quantum system, this procedure seeks to obtain the parameters of a given Hamiltonian model and description of noise sources. Standard techniques for Hamiltonian learning require careful design of queries and $O(ε^{-2})$ queries in achieving learning error $ε$ due to the standard quantum limit. With the goal of efficiently and accurately estimating the Hamiltonian parameters within learning error $ε$ through minimal queries, we introduce an active learner that is given an initial set of training examples and the ability to interactively query the quantum system to generate new training data. We formally specify and experimentally assess the performance of this Hamiltonian active learning (HAL) algorithm for learning the six parameters of a two-qubit cross-resonance Hamiltonian on four different superconducting IBM Quantum devices. Compared with standard techniques for the same problem and a specified learning error, HAL achieves up to a $99.8\%$ reduction in queries required, and a $99.1\%$ reduction over the comparable non-adaptive learning algorithm. Moreover, with access to prior information on a subset of Hamiltonian parameters and given the ability to select queries with linearly (or exponentially) longer system interaction times during learning, HAL can exceed the standard quantum limit and achieve Heisenberg (or super-Heisenberg) limited convergence rates during learning.
In most error correction coding (ECC) frameworks, the typical error metric is the bit error rate (BER) which measures the number of bit errors. For this metric, the positions of the bits are not relevant to the decoding, and in many noise models, not relevant to the BER either. In many applications this is unsatisfactory as typically all bits are not equal and have different significance. We consider the problem of bit error correction and mitigation where bits in different positions have different importance. For error correction, we look at ECC from a Bayesian perspective and introduce Bayes estimators with general loss functions to take into account the bit significance. We propose ECC schemes that optimize this error metric. As the problem is highly nonlinear, traditional ECC construction techniques are not applicable. Using exhaustive search is cost prohibitive, and thus we use iterative improvement search techniques to find good codebooks. We optimize both general codebooks and linear codes. We provide numerical experiments to show that they can be superior to classical linear block codes such as Hamming codes and decoding methods such as minimum distance decoding. For error mitigation, we study the case where ECC is not possible or not desirable, but significance aware encoding of information is still beneficial in reducing the average error. We propose a novel number presentation format suitable for emerging storage media where the noise magnitude is unknown and possibly large and show that it has lower mean error than the traditional number format.
Sparse linear system solvers are computationally expensive kernels that lie at the heart of numerous applications. This paper proposes a flexible preconditioning framework to substantially reduce the time and energy requirements of this task by utilizing a hybrid architecture that combines conventional digital microprocessors with analog crossbar array accelerators. Our analysis and experiments with a simulator for analog hardware demonstrate that an order of magnitude speedup is readily attainable without much impact on convergence, despite the noise in analog computations.
Schur's inequality for the sum of products of the differences of real numbers states that for $x,y,z,t\geq 0$, $x^t(x-y)(x-z) + y^t(y-z)(y-x) + z^t(z-x)(z-y) \geq 0$. In this paper we study a generalization of this inequality to more terms, more general functions of the variables and algebraic structures such as vectors and Hermitian matrices.
Stochastic computing has a long history as an alternative method of performing arithmetic on a computer. While it can be considered an unbiased estimator of real numbers, it has a variance and MSE on the order of $Ω(\frac{1}{N})$. On the other hand, deterministic variants of stochastic computing remove the stochastic aspect, but cannot approximate arbitrary real numbers with arbitrary precision and are biased estimators. However, they have an asymptotically superior MSE on the order of $O(\frac{1}{N^2})$. Recent results in deep learning with stochastic rounding suggest that the bias in the rounding can degrade performance. We proposed an alternative framework, called dither computing, that combines aspects of stochastic computing and its deterministic variants and that can perform computing with similar efficiency, is unbiased, and with a variance and MSE also on the optimal order of $Θ(\frac{1}{N^2})$. We also show that it can be beneficial in stochastic rounding applications as well. We provide implementation details and give experimental results to comparatively show the benefits of the proposed scheme.
We consider a new form of reinforcement learning (RL) that is based on opportunities to directly learn the optimal control policy and a general Markov decision process (MDP) framework devised to support these opportunities. Derivations of general classes of our control-based RL methods are presented, together with forms of exploration and exploitation in learning and applying the optimal control policy over time. Our general MDP framework extends the classical Bellman operator and optimality criteria by generalizing the definition and scope of a policy for any given state. We establish the convergence and optimality-both in general and within various control paradigms (e.g., piecewise linear control policies)-of our control-based methods through this general MDP framework, including convergence of $Q$-learning within the context of our MDP framework. Our empirical results demonstrate and quantify the significant benefits of our approach.
We study synchronization and consensus in a group of dynamical systems coupled via multiple directed networks. We show that even though the coupling in a single network may not be sufficient to synchronize the systems, combination of multiple networks can contribute to synchronization. We illustrate how the effectiveness of a collection of networks to synchronize the coupled systems depends on the graph topology. In particular, we show that if the graph sum is a directed graph whose reversal contains a spanning directed tree, then the network synchronizes if the coupling is strong enough. This is intuitive as there is a root node that influence every other node via a set of edges where each edge in the set is in one of the networks.
We consider the problem of finding the set of permutations $r_j$ of $\{1,\cdots , n\}$ such that $\sum_{i=1}^n \prod_{j=1}^k r_j(i)$ is maximized or minimized. While the set of permutations maximizing this value are easily determined, finding the set of permutations minimizing this value appears to be an open problem. We show values of $k$ and $n$ for which an explicit solution exists and comment on computational issues in determining the general problem. We also look at the dual problem of finding the permutations such that $\prod_{i=1}^n \sum_{j=1}^k r_j(i)$ is maximized or minimized. As part of this study we also look at a variant of a rearrangement inequality.
We consider the use of look-up tables (LUT) to simplify the hardware implementation of a deep learning network for inferencing after weights have been successfully trained. The use of LUT replaces the matrix multiply and add operations with a small number of LUTs and addition operations resulting in a completely multiplier-less implementation. We compare the different tradeoffs of this approach in terms of accuracy versus LUT size and the number of operations and show that similar performance can be obtained with a comparable memory footprint as a full precision deep neural network, but without the use of any multipliers. We illustrate this with several architectures such as MLP and CNN.
We consider a new family of operators for reinforcement learning with the goal of alleviating the negative effects and becoming more robust to approximation or estimation errors. Various theoretical results are established, which include showing on a sample path basis that our family of operators preserve optimality and increase the action gap. Our empirical results illustrate the strong benefits of our family of operators, significantly outperforming the classical Bellman operator and recently proposed operators.
We consider a general framework for reducing the number of trainable model parameters in deep learning networks by decomposing linear operators as a product of sums of simpler linear operators. Recently proposed deep learning architectures such as CNN, KFC, Dilated CNN, etc. are all subsumed in this framework and we illustrate other types of neural network architectures within this framework. We show that good accuracy on MNIST and Fashion MNIST can be obtained using a relatively small number of trainable parameters. In addition, since implementation of the convolutional layer is resource-heavy, we consider an approach in the transform domain that obviates the need for convolutional layers. One of the advantages of this general framework over prior approaches is that the number of trainable parameters is not fixed and can be varied arbitrarily. In particular, we illustrate the tradeoff of varying the number of trainable variables and the corresponding error rate. As an example, by using this decomposition on a reference CNN architecture for MNIST with over 3x10^6 trainable parameters, we are able to obtain an accuracy of 98.44% using only 3554 trainable parameters.
In this short note, we show a simple characterization of integers that reach records for a sequence described by adding binary strings to runs of 1's and 0's in a binary representation. In particular, we show that this set does not depend on the added strings as long as they are nonempty and of the same length.
We explore the possibility of using machine learning to identify interesting mathematical structures by using certain quantities that serve as fingerprints. In particular, we extract features from integer sequences using two empirical laws: Benford's law and Taylor's law and experiment with various classifiers to identify whether a sequence is, for example, nice, important, multiplicative, easy to compute or related to primes or palindromes.
The optimal control of epidemic-like stochastic processes is important both historically and for emerging applications today, where it can be especially important to include time-varying parameters that impact viral epidemic-like propagation. We connect the control of such stochastic processes with time-varying behavior to the stochastic shortest path problem and obtain solutions for various cost functions. Then, under a mean-field scaling, this general class of stochastic processes is shown to converge to a corresponding dynamical system. We analogously establish that the optimal control of this class of processes converges to the optimal control of the limiting dynamical system. Consequently, we study the optimal control of the dynamical system where the comparison of both controlled systems renders various important mathematical properties of interest.
The study of density-dependent stochastic population processes is important from a historical perspective as well as from the perspective of a number of existing and emerging applications today. In more recent applications of these processes, it can be especially important to include time-varying parameters for the rates that impact the density-dependent population structures and behaviors. Under a mean-field scaling, we show that such density-dependent stochastic population processes with time-varying behavior converge to a corresponding dynamical system. We analogously establish that the optimal control of such density-dependent stochastic population processes converges to the optimal control of the limiting dynamical system. An analysis of both the dynamical system and its optimal control renders various important mathematical properties of interest.
We consider the optimal allocation of generic resources among multiple generic entities of interest over a finite planning horizon, where each entity generates stochastic returns as a function of its resource allocation during each period. The main objective is to maximize the expected return while at the same time managing risk to an acceptable level for each period. We devise a general solution framework and establish how to obtain the optimal dynamic resource allocation.