For adversary's strategy defined in This is the desired result in the lemma. Rearranging the above inequality will yield us desired result. On the other hand, we can also upper bound the above conditional mutual information. Putting together the pieces yields our result. We first prove the result for point error, the result of function error can be achieved by a Jensen's inequality (please see the end of the proof). Convexity is maintained by the maximum operator over two convex functions.

We propose a novel classification framework grounded in symbolic dynamics and data compression using chaotic maps. The core idea is to model each class by generating symbolic sequences from thresholded real-valued training data, which are then evolved through a one-dimensional chaotic map. For each class, we compute the transition probabilities of symbolic patterns (e.g., `00', `01', `10', and `11' for the second return map) and aggregate these statistics to form a class-specific probabilistic model. During testing phase, the test data are thresholded and symbolized, and then encoded using the class-wise symbolic statistics via back iteration, a dynamical reconstruction technique. The predicted label corresponds to the class yielding the shortest compressed representation, signifying the most efficient symbolic encoding under its respective chaotic model. This approach fuses concepts from dynamical systems, symbolic representations, and compression-based learning. We evaluate the proposed method: \emph{ChaosComp} on both synthetic and real-world datasets, demonstrating competitive performance compared to traditional machine learning algorithms (e.g., macro F1-scores for the proposed method on Breast Cancer Wisconsin = 0.9531, Seeds = 0.9475, Iris = 0.8469 etc.). Rather than aiming for state-of-the-art performance, the goal of this research is to reinterpret the classification problem through the lens of dynamical systems and compression, which are foundational perspectives in learning theory and information processing.

Dynamic imaging involves the reconstruction of a spatio-temporal object at all times using its undersampled measurements. In particular, in dynamic computed tomography (dCT), only a single projection at one view angle is available at a time, making the inverse problem very challenging. Moreover, ground-truth dynamic data is usually either unavailable or too scarce to be used for supervised learning techniques. To tackle this problem, we propose RSR-NF, which uses a neural field (NF) to represent the dynamic object and, using the Regularization-by-Denoising (RED) framework, incorporates an additional static deep spatial prior into a variational formulation via a learned restoration operator. We use an ADMM-based algorithm with variable splitting to efficiently optimize the variational objective. We compare RSR-NF to three alternatives: NF with only temporal regularization; a recent method combining a partially-separable low-rank representation with RED using a denoiser pretrained on static data; and a deep-image prior-based model. The first comparison demonstrates the reconstruction improvements achieved by combining the NF representation with static restoration priors, whereas the other two demonstrate the improvement over state-of-the art techniques for dCT.

We analyze different approaches to differential-algebraic equations with attention to the implemented rank conditions of various matrix functions. These conditions are apparently very different and certain rank drops in some matrix functions actually indicate a critical solution behavior. We look for common ground by considering various index and regularity notions from literature generalizing the Kronecker index of regular matrix pencils. In detail, starting from the most transparent reduction framework, we work out a comprehensive regularity concept with canonical characteristic values applicable across all frameworks and prove the equivalence of thirteen distinct definitions of regularity. This makes it possible to use the findings of all these concepts together. Additionally, we show why not only the index but also these canonical characteristic values are crucial to describe the properties of the DAE.

Given a pair of multivariate time-series data of the same length and dimensions, an approach is proposed to select variables and time intervals where the two series are significantly different. In applications where one time series is an output from a computationally expensive simulator, the approach may be used for validating the simulator against real data, for comparing the outputs of two simulators, and for validating a machine learning-based emulator against the simulator. With the proposed approach, the entire time interval is split into multiple subintervals, and on each subinterval, the two sample sets are compared to select variables that distinguish their distributions and a two-sample test is performed. The validity and limitations of the proposed approach are investigated in synthetic data experiments. Its usefulness is demonstrated in an application with a particle-based fluid simulator, where a deep neural network model is compared against the simulator, and in an application with a microscopic traffic simulator, where the effects of changing the simulator's parameters on traffic flows are analysed.

While deep learning has been successfully applied to the data-driven classification of anomalous diffusion mechanisms, how the algorithm achieves the feat still remains a mystery. In this study, we use a well-known technique aimed at achieving explainable AI, namely the Gradient-weighted Class Activation Map (Grad-CAM), to investigate how deep learning (implemented by ResNets) recognizes the distinctive features of a particular anomalous diffusion model from the raw trajectory data. Our results show that Grad-CAM reveals the portions of the trajectory that hold crucial information about the underlying mechanism of anomalous diffusion, which can be utilized to enhance the robustness of the trained classifier against the measurement noise. Moreover, we observe that deep learning distills unique statistical characteristics of different diffusion mechanisms at various spatiotemporal scales, with larger-scale (smaller-scale) features identified at higher (lower) layers.

The Poisson process, especially the nonhomogeneous Poisson process (NHPP), is an essentially important counting process with numerous real-world applications. Up to date, almost all works in the literature have been on the estimation of NHPPs with infinite data using non-data driven binning methods. In this paper, we formulate the problem of estimation of NHPPs from finite and limited data as a learning generalization problem. We mathematically show that while binning methods are essential for the estimation of NHPPs, they pose a threat of overfitting when the amount of data is limited. We propose a framework for regularized learning of NHPPs with two new adaptive and data-driven binning methods that help to remove the ad-hoc tuning of binning parameters. Our methods are experimentally tested on synthetic and real-world datasets and the results show their effectiveness.

We revisit the original approach of using deep learning and neural networks to solve differential equations by incorporating the knowledge of the equation. This is done by adding a dedicated term to the loss function during the optimization procedure in the training process. The so-called physics-informed neural networks (PINNs) are tested on a variety of academic ordinary differential equations in order to highlight the benefits and drawbacks of this approach with respect to standard integration methods. We focus on the possibility to use the least possible amount of data into the training process. The principles of PINNs for solving differential equations by enforcing physical laws via penalizing terms are reviewed. A tutorial on a simple equation model illustrates how to put into practice the method for ordinary differential equations. Benchmark tests show that a very small amount of training data is sufficient to predict the solution when the non linearity of the problem is weak. However, this is not the case in strongly non linear problems where a priori knowledge of training data over some partial or the whole time integration interval is necessary.

Let two sequences of eventualities x (signifying the sequence, x0,x1, x2,...,xn-1) and y (signifying the sequence, y0, y1, y2,..,yn-1) both recur over the same time interval and it is required to determine whether or not a subinterval exists within the said interval which is a common subinterval of the intervals of occurrence of xp and yq. This paper presents two algorithms for solving the problem. the first explores an arbitrary cycle of the double recurrence for the existence of such an interval. its worst case running time is quadratic. The other algorithm is based on the novel notion of gcd-partitions and has a linear worst case running time. If the eventuality sequence pair (W,z) is a gcd-partition for the double recurrence (x, y),then, from a certain property of gcd-partitions, within any cycle of the double recurrence, there exists r and s such that intervals of occurrence of xp and yq are non-disjoint with the interval of co-occurrence of wr and zs. As such, a coincidence between xp and yq occurs within a cycle of the double recurrence if and only if such r and s exist so that the interval of co-occurrence of wr and zs shares a common interval with the common interval of occurrences of xp and yq. The algorithm systematically reduces the number of wr and zs pairs to be explored in the process of finding the existence of the coincidence.