This paper explores the enhancement of solution diversity in evolutionary algorithms (EAs) for the maximum matching problem, concentrating on complete bipartite graphs and paths. We adopt binary string encoding for matchings and use Hamming distance to measure diversity, aiming for its maximization. Our study centers on the $(\mu+1)$-EA and $2P-EA_D$, which are applied to optimize diversity. We provide a rigorous theoretical and empirical analysis of these algorithms. For complete bipartite graphs, our runtime analysis shows that, with a reasonably small $\mu$, the $(\mu+1)$-EA achieves maximal diversity with an expected runtime of $O(\mu^2 m^4 \log(m))$ for the small gap case (where the population size $\mu$ is less than the difference in the sizes of the bipartite partitions) and $O(\mu^2 m^2 \log(m))$ otherwise. For paths, we establish an upper runtime bound of $O(\mu^3 m^3)$. The $2P-EA_D$ displays stronger performance, with bounds of $O(\mu^2 m^2 \log(m))$ for the small gap case, $O(\mu^2 n^2 \log(n))$ otherwise, and $O(\mu^3 m^2)$ for paths. Here, $n$ represents the total number of vertices and $m$ the number of edges. Our empirical studies, which examine the scaling behavior with respect to $m$ and $\mu$, complement these theoretical insights and suggest potential for further refinement of the runtime bounds.
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This paper discusses one of the most challenging practical engineering problems in speaker recognition systems - the version control of models and user profiles. A typical speaker recognition system consists of two stages: the enrollment stage, where a profile is generated from user-provided enrollment audio; and the runtime stage, where the voice identity of the runtime audio is compared against the stored profiles. As technology advances, the speaker recognition system needs to be updated for better performance. However, if the stored user profiles are not updated accordingly, version mismatch will result in meaningless recognition results. In this paper, we describe different version control strategies for speaker recognition systems that had been carefully studied at Google from years of engineering practice. These strategies are categorized into three groups according to how they are deployed in the production environment: device-side deployment, server-side deployment, and hybrid deployment. To compare different strategies with quantitative metrics under various network configurations, we present SpeakerVerSim, an easily-extensible Python-based simulation framework for different server-side deployment strategies of speaker recognition systems.
This manuscript presents a novel method for discovering effective connectivity between specified pairs of nodes in a high-dimensional network of time series. To accurately perform Granger causality analysis from the first node to the second node, it is essential to eliminate the influence of all other nodes within the network. The approach proposed is to create a low-dimensional representation of all other nodes in the network using frequency-domain-based dynamic principal component analysis (spectral DPCA). The resulting scores are subsequently removed from the first and second nodes of interest, thus eliminating the confounding effect of other nodes within the high-dimensional network. To conduct hypothesis testing on Granger causality, we propose a permutation-based causality test. This test enhances the accuracy of our findings when the error structures are non-Gaussian. The approach has been validated in extensive simulation studies, which demonstrate the efficacy of the methodology as a tool for causality analysis in complex time series networks. The proposed methodology has also been demonstrated to be both expedient and viable on real datasets, with particular success observed on multichannel EEG networks.
The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a "uniform" bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.
This paper proposes a verification method for sparse linear systems $Ax=b$ with general and nonsingular coefficients. A verification method produces the error bound for a given approximate solution. Conventional methods use one of two approaches. One approach is to verify the computed solution of the normal equation $A^TAx=A^Tb$ by exploiting symmetric and positive definiteness; however, the condition number of $A^TA$ is the square of that for $A$. The other approach uses an approximate inverse matrix of the coefficient; however, the approximate inverse may be dense even if $A$ is sparse. Here, we propose a method for the verification of solutions of sparse linear systems based on $LDL^T$ decomposition. The proposed method can reduce the fill-in and is applicable to many problems. Moreover, an efficient iterative refinement method is proposed for obtaining accurate solutions.
This paper deals with efficient numerical methods for computing the action of the generating function of Bernoulli polynomials, say $q(\tau,w)$, on a typically large sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of $q(\tau,w)$ have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms.
Mixed linear regression is a well-studied problem in parametric statistics and machine learning. Given a set of samples, tuples of covariates and labels, the task of mixed linear regression is to find a small list of linear relationships that best fit the samples. Usually it is assumed that the label is generated stochastically by randomly selecting one of two or more linear functions, applying this chosen function to the covariates, and potentially introducing noise to the result. In that situation, the objective is to estimate the ground-truth linear functions up to some parameter error. The popular expectation maximization (EM) and alternating minimization (AM) algorithms have been previously analyzed for this. In this paper, we consider the more general problem of agnostic learning of mixed linear regression from samples, without such generative models. In particular, we show that the AM and EM algorithms, under standard conditions of separability and good initialization, lead to agnostic learning in mixed linear regression by converging to the population loss minimizers, for suitably defined loss functions. In some sense, this shows the strength of AM and EM algorithms that converges to ``optimal solutions'' even in the absence of realizable generative models.
In the field of crowd counting research, many recent deep learning based methods have demonstrated robust capabilities for accurately estimating crowd sizes. However, the enhancement in their performance often arises from an increase in the complexity of the model structure. This paper discusses how to construct high-performance crowd counting models using only simple structures. We proposes the Fuss-Free Network (FFNet) that is characterized by its simple and efficieny structure, consisting of only a backbone network and a multi-scale feature fusion structure. The multi-scale feature fusion structure is a simple structure consisting of three branches, each only equipped with a focus transition module, and combines the features from these branches through the concatenation operation. Our proposed crowd counting model is trained and evaluated on four widely used public datasets, and it achieves accuracy that is comparable to that of existing complex models. Furthermore, we conduct a comprehensive evaluation by replacing the existing backbones of various models such as FFNet and CCTrans with different networks, including MobileNet-v3, ConvNeXt-Tiny, and Swin-Transformer-Small. The experimental results further indicate that excellent crowd counting performance can be achieved with the simplied structure proposed by us.
This paper studies the uncertainty set estimation of system parameters of linear dynamical systems with bounded disturbances, which is motivated by robust (adaptive) constrained control. Departing from the confidence bounds of least square estimation from the machine-learning literature, this paper focuses on a method commonly used in (robust constrained) control literature: set membership estimation (SME). SME tends to enjoy better empirical performance than LSE's confidence bounds when the system disturbances are bounded. However, the theoretical guarantees of SME are not fully addressed even for i.i.d. bounded disturbances. In the literature, SME's convergence has been proved for general convex supports of the disturbances, but SME's convergence rate assumes a special type of disturbance support: $l_\infty$ ball. The main contribution of this paper is relaxing the assumption on the disturbance support and establishing the convergence rates of SME for general convex supports, which closes the gap on the applicability of the convergence and convergence rates results. Numerical experiments on SME and LSE's confidence bounds are also provided for different disturbance supports.
This paper studies a beam tracking problem in which an access point (AP), in collaboration with a reconfigurable intelligent surface (RIS), dynamically adjusts its downlink beamformers and the reflection pattern at the RIS in order to maintain reliable communications with multiple mobile user equipments (UEs). Specifically, the mobile UEs send uplink pilots to the AP periodically during the channel sensing intervals, the AP then adaptively configures the beamformers and the RIS reflection coefficients for subsequent data transmission based on the received pilots. This is an active sensing problem, because channel sensing involves configuring the RIS coefficients during the pilot stage and the optimal sensing strategy should exploit the trajectory of channel state information (CSI) from previously received pilots. Analytical solution to such an active sensing problem is very challenging. In this paper, we propose a deep learning framework utilizing a recurrent neural network (RNN) to automatically summarize the time-varying CSI obtained from the periodically received pilots into state vectors. These state vectors are then mapped to the AP beamformers and RIS reflection coefficients for subsequent downlink data transmissions, as well as the RIS reflection coefficients for the next round of uplink channel sensing. The mappings from the state vectors to the downlink beamformers and the RIS reflection coefficients for both channel sensing and downlink data transmission are performed using graph neural networks (GNNs) to account for the interference among the UEs. Simulations demonstrate significant and interpretable performance improvement of the proposed approach over the existing data-driven methods with nonadaptive channel sensing schemes.
The remarkable instruction-following capability of large language models (LLMs) has sparked a growing interest in automatically finding good prompts, i.e., prompt optimization. Most existing works follow the scheme of selecting from a pre-generated pool of candidate prompts. However, these designs mainly focus on the generation strategy, while limited attention has been paid to the selection method. Especially, the cost incurred during the selection (e.g., accessing LLM and evaluating the responses) is rarely explicitly considered. To overcome this limitation, this work provides a principled framework, TRIPLE, to efficiently perform prompt selection under an explicit budget constraint. TRIPLE is built on a novel connection established between prompt optimization and fixed-budget best arm identification (BAI-FB) in multi-armed bandits (MAB); thus, it is capable of leveraging the rich toolbox from BAI-FB systematically and also incorporating unique characteristics of prompt optimization. Extensive experiments on multiple well-adopted tasks using various LLMs demonstrate the remarkable performance improvement of TRIPLE over baselines while satisfying the limited budget constraints. As an extension, variants of TRIPLE are proposed to efficiently select examples for few-shot prompts, also achieving superior empirical performance.
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