In many real world problems, control decisions have to be made with limited information. The controller may have no a priori (or even posteriori) data on the nonlinear system, except from a limited number of points that are obtained over time. This is either due to high cost of observation or the highly non-stationary nature of the system. The resulting conflict between information collection (identification, exploration) and control (optimization, exploitation) necessitates an active learning approach for iteratively selecting the control actions which concurrently provide the data points for system identification. This paper presents a dual control approach where the information acquired at each control step is quantified using the entropy measure from information theory and serves as the training input to a state-of-the-art Gaussian process regression (Bayesian learning) method. The explicit quantification of the information obtained from each data point allows for iterative optimization of both identification and control objectives. The approach developed is illustrated with two examples: control of logistic map as a chaotic system and position control of a cart with inverted pendulum.

We consider decentralized restless multi-armed bandit problems with unknown dynamics and multiple players. The reward state of each arm transits according to an unknown Markovian rule when it is played and evolves according to an arbitrary unknown random process when it is passive. Players activating the same arm at the same time collide and suffer from reward loss. The objective is to maximize the long-term reward by designing a decentralized arm selection policy to address unknown reward models and collisions among players. A decentralized policy is constructed that achieves a regret with logarithmic order when an arbitrary nontrivial bound on certain system parameters is known. When no knowledge about the system is available, we extend the policy to achieve a regret arbitrarily close to the logarithmic order. The result finds applications in communication networks, financial investment, and industrial engineering.

Recent years have witnessed the popularity of using rank minimization as a regularizer for various signal processing and machine learning problems. As rank minimization problems are often converted to nuclear norm minimization (NNM) problems, they have to be solved iteratively and each iteration requires computing a singular value decomposition (SVD). Therefore, their solution suffers from the high computation cost of multiple SVDs. To relieve this issue, we propose using the block Lanczos method to compute the partial SVDs, where the principal singular subspaces obtained in the previous iteration are used to start the block Lanczos procedure. To avoid the expensive reorthogonalization in the Lanczos procedure, the block Lanczos procedure is performed for only a few steps. Our block Lanczos with warm start (BLWS) technique can be adopted by different algorithms that solve NNM problems. We present numerical results on applying BLWS to Robust PCA and Matrix Completion problems. Experimental results show that our BLWS technique usually accelerates its host algorithm by at least two to three times.

Suppose a given observation matrix can be decomposed as the sum of a low-rank matrix and a sparse matrix (outliers), and the goal is to recover these individual components from the observed sum. Such additive decompositions have applications in a variety of numerical problems including system identification, latent variable graphical modeling, and principal components analysis. We study conditions under which recovering such a decomposition is possible via a combination of $\ell_1$ norm and trace norm minimization. We are specifically interested in the question of how many outliers are allowed so that convex programming can still achieve accurate recovery, and we obtain stronger recovery guarantees than previous studies. Moreover, we do not assume that the spatial pattern of outliers is random, which stands in contrast to related analyses under such assumptions via matrix completion.

Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and partially observed signals respectively. We combine the distance preserving measurements of compressed sensing and matrix completion with the power of robust spectral clustering. Our analysis provides rigorous bounds on how small errors in the affinity matrix can affect the spectral coordinates and clusterability. This work generalizes the current perturbation results of two-class spectral clustering to incorporate multi-class clustering with k eigenvectors. We thoroughly track how small perturbation from using compressed sensing and matrix completion affect the affinity matrix and in succession the spectral coordinates. These perturbation results for multi-class clustering require an eigengap between the kth and (k+1)th eigenvalues of the affinity matrix, which naturally occurs in data with k well-defined clusters. Our theoretical guarantees are complemented with numerical results along with a number of examples of the unsupervised organization and clustering of image data.

We present in this paper first-order alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most $O(1/ε)$ iterations to obtain an $ε$-optimal solution, while our accelerated (i.e., fast) versions of them require at most $O(1/\sqrtε)$ iterations, with little change in the computational effort required at each iteration. For both types of methods, we present one algorithm that requires both functions to be smooth with Lipschitz continuous gradients and one algorithm that needs only one of the functions to be so. Algorithms in this paper are Gauss-Seidel type methods, in contrast to the ones proposed by Goldfarb and Ma in [21] where the algorithms are Jacobi type methods. Numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms.

This paper illustrates the application of recent research in region-of-attraction analysis for nonlinear hybrid limit cycles. Three example systems are analyzed in detail: the van der Pol oscillator, the "rimless wheel", and the "compass gait", the latter two being simplified models of underactuated walking robots. The method used involves decomposition of the dynamics about the target cycle into tangential and transverse components, and a search for a Lyapunov function in the transverse dynamics using sum-of-squares analysis (semidefinite programming). Each example illuminates different aspects of the procedure, including optimization of transversal surfaces, the handling of impact maps, optimization of the Lyapunov function, and orbitally-stabilizing control design.

The famous Perona-Malik (P-M) equation which was at first introduced for image restoration has been solved via various numerical methods. In this paper we will solve it for the first time via applying a new numerical method called the Variational Iteration Method (VIM) and the correspondent approximated solutions will be obtained for the P-M equation with regards to relevant error analysis. Through implementation of our algorithm we will access some effective results which are deserved to be considered as worthy as the other solutions issued by the other methods.

Euclidean norm calculations arise frequently in scientific and engineering applications. Several approximations for this norm with differing complexity and accuracy have been proposed in the literature. Earlier approaches were based on minimizing the maximum error. Recently, Seol and Cheun proposed an approximation based on minimizing the average error. In this paper, we first examine these approximations in detail, show that they fit into a single mathematical formulation, and compare their average and maximum errors. We then show that the maximum errors given by Seol and Cheun are significantly optimistic.

In this paper, we propose a systematic solution to the problem of scheduling delay-sensitive media data for transmission over time-varying wireless channels. We first formulate the dynamic scheduling problem as a Markov decision process (MDP) that explicitly considers the users' heterogeneous multimedia data characteristics (e.g. delay deadlines, distortion impacts and dependencies etc.) and time-varying channel conditions, which are not simultaneously considered in state-of-the-art packet scheduling algorithms. This formulation allows us to perform foresighted decisions to schedule multiple data units for transmission at each time in order to optimize the long-term utilities of the multimedia applications. The heterogeneity of the media data enables us to express the transmission priorities between the different data units as a priority graph, which is a directed acyclic graph (DAG). This priority graph provides us with an elegant structure to decompose the multi-data unit foresighted decision at each time into multiple single-data unit foresighted decisions which can be performed sequentially, from the high priority data units to the low priority data units, thereby significantly reducing the computation complexity. When the statistical knowledge of the multimedia data characteristics and channel conditions is unknown a priori, we develop a low-complexity online learning algorithm to update the value functions which capture the impact of the current decision on the future utility. The simulation results show that the proposed solution significantly outperforms existing state-of-the-art scheduling solutions.

Non-negative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. We interpret the factorization in a new way and use it to generate missing attributes from test data. We provide a joint optimization scheme for the missing attributes as well as the NMF factors. We prove the monotonic convergence of our algorithms. We present classification results for cases with missing attributes.

We consider the application of the polyharmonic subdivision wavelets (of Daubechies type) to Image Processing, in particular to Astronomical Images. The results show an essential advantage over some standard multivariate wavelets and a potential for better compression.

In this article we introduce a broad family of adaptive, linear time-frequency representations termed superposition frames, and show that they admit desirable fast overlap-add reconstruction properties akin to standard short-time Fourier techniques. This approach stands in contrast to many adaptive time-frequency representations in the extant literature, which, while more flexible than standard fixed-resolution approaches, typically fail to provide efficient reconstruction and often lack the regular structure necessary for precise frame-theoretic analysis. Our main technical contributions come through the development of properties which ensure that this construction provides for a numerically stable, invertible signal representation. Our primary algorithmic contributions come via the introduction and discussion of specific signal adaptation criteria in deterministic and stochastic settings, based respectively on time-frequency concentration and nonstationarity detection. We conclude with a short speech enhancement example that serves to highlight potential applications of our approach.

Mixed dictionaries generated by cosine and B-spline functions are considered. It is shown that, by highly nonlinear approaches such as Orthogonal Matching Pursuit, the discrete version of the proposed dictionaries yields a significant gain in the sparsity of an image representation.

In this paper, we examine the CE method in the broad context of Monte Carlo Optimization (MCO) and Parametric Learning (PL), a type of machine learning. A well-known overarching principle used to improve the performance of many PL algorithms is the bias-variance tradeoff. This tradeoff has been used to improve PL algorithms ranging from Monte Carlo estimation of integrals, to linear estimation, to general statistical estimation. Moreover, as described by, MCO is very closely related to PL. Owing to this similarity, the bias-variance tradeoff affects MCO performance, just as it does PL performance. In this article, we exploit the bias-variance tradeoff to enhance the performance of MCO algorithms. We use the technique of cross-validation, a technique based on the bias-variance tradeoff, to significantly improve the performance of the Cross Entropy (CE) method, which is an MCO algorithm. In previous work we have confirmed that other PL techniques improve the perfomance of other MCO algorithms. We conclude that the many techniques pioneered in PL could be investigated as ways to improve MCO algorithms in general, and the CE method in particular.

Artificial Neural Network (ANN) is used as numerical methode in solving modified Nonlinear Schroedinger (NLS) equation with Dispersion Managed System (DMS) for jitter analysis. We take the optical axis z and the time t as input, and then some relevant values such as the change of position and the center frequency of the pulse, and further the mean square time of incoming pulse which are needed for jitter analysis. It shows that ANN yields numerical solutions which are adaptive with respect to the numerical errors and also verifies the previous numerical results using conventional numerical method. Our result indicates that DMS can minimize the timing jitter induced by some amplifiers.

We show that information on the first server influences the expected total number of games and margin in a tennis match under the standard assumption that each player's serve point win probability remains constant, and identify the exact regions, in terms of these probabilities, in which this effect is non-negligible. We confirm numerically that this effect is bounded by at most $0.4$ games at both the set and match level. This translates, for example, to roughly a $9$ percent shift in the probability that a match exceeds $19.5$ games when the players' serve point win probabilities differ by $10$ percent. We complement the analysis with an empirical comparison on professional match data, illustrating the adequacy of the constant-probability assumption for modelling the total number of games.

Superconductivity has recently been observed in moiré transition-metal dichalcogenide bilayers. Here, we investigate the superconducting state in twisted WSe$_2$ using two complementary theoretical approaches. The first is based on the negative $U$-Hubbard model and represents a relatively conventional pairing scenario, in which strong electron-electron repulsion does not directly affect the paired state and an isotropic $s$-$wave$ gap emerges. The second approach employs the $t$-$J$-$U$ model, allowing for unconventional gap symmetries and incorporating strong correlation effects via substantial renormalization induced by Coulomb repulsion. We compare the key properties of the superconducting states obtained within these two frameworks and discuss their implications in light of available experimental observations.

The centre of the Milky Way is occupied by a nuclear star cluster that contains the supermassive black hole Sgr A*. The cluster is embedded in the larger surrounding nuclear stellar disc. These three components dominate the mass budget of the Galactic centre at different radial scales. The mass distribution of the Galactic centre has been studied extensively using observations of individual bright stars and various dynamical modelling approaches. The situation differs for external galaxies, where observations are often limited to the integrated line-of-sight kinematics. For such systems, triaxial orbit-based dynamical modelling has become a standard method of deriving mass distributions and stellar orbit distributions. We aim to apply and test this method on the Galactic centre. We extracted stellar line-of-sight kinematic maps of the inner ~3 pc x 66 pc region of the Galactic centre. We used the DYNAMITE code, which calculates an orbit library in a given gravitational potential and computes model kinematic maps. These maps were then compared to the observed kinematic maps, and the gravitational potential and orbit distribution of the Galactic centre were constrained. We recover the correct mass of Sgr A*, and our stellar mass distributions are in agreement with the literature, albeit with larger uncertainties. The stellar structures are at most mildly triaxial and close to oblate. The stellar orbit distribution in the inner region is dominated by dynamically warm and hot orbits. At larger scales, dynamically cold -- highly rotating -- orbits have the largest weights. The dominance of hot and warm orbits is a consequence of short dynamical timescales in the inner Galactic centre, causing dynamical heating. The presence of cold orbits at large radii may be explained by the longer heating timescales in this region, and by the stars in the outer nuclear stellar disc being younger.[abridged]

Classroom dynamics depend on various elements that influence teaching performance and learning activities. A key challenge is to determine the most effective seating plan, where students will seat in a specific classroom setting to achieve the best learning environment. This paper introduces the Student Seat Allocation Problem (SSAP) for strategically organizing student seating in traditional classrooms to minimize interpersonal conflicts. We propose a mathematical model and an Iterated Local Search (ILS) heuristic to solve the SSAP. Computational experiments demonstrated that ILS outperformed in more complex scenarios when compared to the results obtained by a commercial solver on the introduced mathematical model. ILS was particularly efficient in real and artificial instances that exhibited a higher number of conflicts.

We study the generalized Lam'e equation (GLE) on an elliptic curve $E$ with multiple regular singularities $\mathbf{p} = (p_i)_{i = 1}^r$ of weights $\mathbf{n} = (n_i)_{i = 1}^r$. By analyzing the locus admitting quasi-periodic solutions, we construct two fundamental algebraic curves: (i) The generalized Lam'e curve (GLC), $\mathcal{Y}_{\mathbf{n}, \mathbf{p}}$, which lies in an affine bundle over $\operatorname{Sym}^n E$ for total weight $n:=\sum n_i \in \mathbb{Z}_{\geq 0}$ and parametrizes generalized Hermite--Halphen ansatz solutions. (ii) The log-free curve, $V_{\mathbf{n}, \mathbf{p}}$, a non-complete intersection variety arising when all $n_i \in \frac{1}{2}\mathbb{N}$, which we prove is a reduced curve, confirming a conjecture of Wang. We analyze the GLC as an algebraic family over the pole configuration space. By studying the addition map$$σ\colon \operatorname{Sym}^n E \longrightarrow E,$$where we establish a generically finite, universal degree formula, we show that the geometry of boundary degenerations under pole collisions perfectly mirrors the tensor algebra of $\mathfrak{sl}_2(\mathbb{C})$-modules within the BGG category $\mathcal{O}$. This provides the local structural limits needed to establish the global flatness of the GLC. Furthermore, we develop a framework of twisted isomonodromic deformations and construct $(\mathbf{n}, \mathbf{p})$-deformed pre-modular forms parameterized by twisted monodromy data $(t,s)$. Their vanishing solves the underlying monodromy problem and factorizes along boundary strata, allowing an arbitrary configuration to be continuously deformed down to the classical Lam'e equation. Finally, using an asymptotic scaling technique, we completely solve the Treibich conjecture for $r=2$ symmetric pairs, extend it to $r \leq 4$, and propose a general formula enumerating symmetric finite-gap KdV potentials for all $r$.

A systematic algebraic framework for composing and decomposing logic programs is currently missing, limiting our ability to analyze and construct programs in a modular way. In this paper, we introduce set-like operations for (propositional Horn) logic programs that allow for a structured manipulation of rule bodies. Our main technical result shows that programs can be decomposed into simpler components in such a way that their least model semantics can be reconstructed or approximated from the semantics of these components. In particular, we prove that every minimalist program can be decomposed into Krom programs -- consisting only of rules with at most one body atom -- such that its least model can be computed from the least models of its components. For arbitrary programs, we obtain corresponding approximation results. These results provide a new algebraic perspective on logic programs and lay the groundwork for compositional reasoning and program construction.

Let $ζ$ be Euclidean norm of the degree sequence of a graph normalized by the graph size. We prove that when the vertices of a graph are randomly colored with $s$ colors such that the fraction of vertices in each color class is bounded away from zero, only two asymptotic regimes emerge. If $ζ=o(1)$, then the sizes of the subgraphs induced by the color classes concentrate around their expected values. If $ζ=Θ(1)$, then concentration depends on the color balance: for colorings with persisting imbalance, the total number $M$ of monochromatic edges stays bounded away from its mean with positive probability; otherwise, for vanishing imbalance, $M$ still concentrates. The same dichotomy holds for a broad class of randomly colored random graphs.

High-dimensional time series regressions are often regularized to produce sparse coefficients. We show that temporal aggregation provides a powerful alternative to reduce dimensionality in high-order autoregressions and mixed-frequency regressions. To this end, we propose StarTime (Sparse Tree-based Aggregation for Time Series), a convex penalization method that uses a temporal tree to arrange lags hierarchically from high to low frequency. StarTime then flexibly selects coefficients to be aggregated at possibly varying frequencies, sparse or a combination thereof. We provide new error bounds for StarTime, demonstrate improved estimation accuracy and recovery of aggregation and sparsity in simulations relative to benchmarks, and illustrate StarTime's relevance for financial and macroeconomic applications.

We study optimal auction design when the direction of bidders' deviations is restricted. We show that the optimal revenue when bidders can only underbid their true values cannot exceed the optimal revenue when bidders may freely underbid or overbid. Thus, unidirectional incentive compatibility is sufficient for full incentive compatibility for revenue maximization. We prove this equivalence through linear programming duality in a discrete model, which makes it possible to analyze the feasibility of allocation rules in multi-agent environments.

Hedonic price models are widely used to assess how environmental amenities affect property values, yet methodological guidance for estimating direct price effects remains sparse. We conduct an empirical Monte Carlo simulation to evaluate the performance of traditional and causal machine learning approaches for estimating the direct unmediated price effect of spatially delineated amenities on treated properties (DUET), a conservative lower-bound approximation for welfare changes with direct applications to benefit-cost analysis. Where previous simulations rely on parametric assumptions, we retain the actual data-generating process underlying over 1 million property transactions from upstate New York (1990--2024). By randomly assigning "treatment locations" across iterations we establish a "ground truth" that allows us to precisely measure estimation error. Our results demonstrate that generalized difference-in-differences (DID) regression consistently outperforms baseline DID and two-way fixed effects models across all scenarios. Causal Machine Learning (CML) methods, particularly causal forest DID, achieve comparable performance to generalized DID in most scenarios. In larger samples (above 3,000 treated) increasingly common in contemporary hedonic studies, CML approaches offer substantial advantages when properly specified. Based on empirical simulation results, we provide a set of method-specific best practice recommendations for both traditional regression and causal machine learning approaches.

A principal who offers a contract may renege when her default option is sufficiently attractive. The size of this temptation, which measures her commitment power, is often her private information. This paper asks how contracting outcomes change under this information asymmetry. Disciplining off-path beliefs with the Intuitive Criterion, I find that every type of principal behaves and earns payoffs exactly as if she were commonly known to have the least commitment power. Hidden commitment power is therefore powerless. The result delivers an unambiguous policy lesson on how to mitigate this information asymmetry prior to contracting: only measures that improve the worst case have value. Applied to credit rating, it rationalizes the monotone-partitional structure widely used in practice.

Acoustic feedback limits the maximum gain in hearing aids. In addition to several approaches based on adaptive filtering, recently a deep-neural-network-based feedback cancellation (DFC) approach has been proposed, which is trained via an open-loop framework. Since open-loop-trained DFC (DFC-OL) can become unstable during inference at high gains, in this paper we propose an in-the-loop-trained DFC (DFC-IL) that integrates the DFC directly into the optimisation loop. This allows the model to be exposed to unstable conditions during training. A two-stage training strategy involving pre-training on stable systems and fine-tuning on a wider gain range enables DFC-IL to learn robust howling reduction. Experimental results on measured feedback paths demonstrate that in scenarios with small gains, the proposed DFC-IL performs similarly to DFC-OL, and both exceed the performance of adaptive filters. In scenarios with high amplification gains, DFC-IL clearly outperforms DFC-OL by maintaining system stability.

Pinching antenna systems (PASS) have recently emerged as a promising architecture for flexible indoor wireless communications. However, most existing pinching antenna (PA) array designs for multi-user PASS either offer limited beam adaptation accuracy or require prohibitively high deployment cost. In this paper, we investigate a more practical constrained pinching antenna array (C-PAA)-assisted downlink PASS, where multiple PAs are grouped into a movable array and can be finely adjusted within the array at the wavelength scale. To improve the system spectral efficiency, a sum-rate maximization problem is formulated by jointly considering the array-center position and the fine-grained antenna distribution within the C-PAA. First, the structural properties of the C-PAA are characterized, and an explicit upper bound on the array aperture is derived. Then, tractable approximations for the effective channel gain and the achievable user rate are developed. Furthermore, the optimization problem of the multi-user sum-rate is analyzed, where the system sum-rate function is shown to exhibit a favorable unimodal behavior under practically relevant conditions, which enables an efficient one-dimensional search for the optimal C-PAA position. To further reduce the computational complexity, a closed-form approximate solution for the near-optimal array-center position is derived. Numerical results verify the accuracy of the developed analysis and demonstrate that the proposed C-PAA scheme closely approaches the ideal upper bound and significantly outperforms conventional fixed-spacing and existing PA array benchmarks.

Audio-Visual Event Recognition (AVER) is essential for intelligent urban monitoring systems, where robust multimodal understanding of complex environments is required. This paper proposes a stable hybrid cross-attention fusion framework for audio-visual event recognition in smart urban environments. The proposed architecture combines pretrained Video Masked Autoencoder (VideoMAE) and Audio Spectrogram Transformer (AST) representations with FiLM-based audio conditioning, bidirectional cross-attention fusion, multimodal Transformer encoding, and modality-temporal attention. To improve computational efficiency and training stability, frozen pretrained backbones and cached feature extraction are employed. Extensive experiments on the AVE dataset show that the proposed framework achieves the highest average performance among the evaluated unimodal and multimodal baselines across multiple evaluation metrics, obtaining a best validation accuracy of 91.74% and a test accuracy of 83.85 plus/minus 1.40% over five independent runs. The results indicate that the proposed hybrid fusion strategy effectively captures complementary audio-visual information and provides robust multimodal representation learning for challenging realworld urban monitoring scenarios.