Using derandomization, we provide an upper bound on the compression size of solutions to the graph coloring problem. In general, if solutions to a combinatorial problem exist with high probability and the probability is simple, then there exists a simple solution to the problem. Otherwise the problem instance has high mutual information with the halting problem.
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Deterministic finite automata (DFA) are a classic tool for high throughput matching of regular expressions, both in theory and practice. Due to their high space consumption, extensive research has been devoted to compressed representations of DFAs that still support efficient pattern matching queries. Kumar~et~al.~[SIGCOMM 2006] introduced the \emph{delayed deterministic finite automaton} (\ddfa{}) which exploits the large redundancy between inter-state transitions in the automaton. They showed it to obtain up to two orders of magnitude compression of real-world DFAs, and their work formed the basis of numerous subsequent results. Their algorithm, as well as later algorithms based on their idea, have an inherent quadratic-time bottleneck, as they consider every pair of states to compute the optimal compression. In this work we present a simple, general framework based on locality-sensitive hashing for speeding up these algorithms to achieve sub-quadratic construction times for \ddfa{}s. We apply the framework to speed up several algorithms to near-linear time, and experimentally evaluate their performance on real-world regular expression sets extracted from modern intrusion detection systems. We find an order of magnitude improvement in compression times, with either little or no loss of compression, or even significantly better compression in some cases.
Exoplanet detection by direct imaging is a difficult task: the faint signals from the objects of interest are buried under a spatially structured nuisance component induced by the host star. The exoplanet signals can only be identified when combining several observations with dedicated detection algorithms. In contrast to most of existing methods, we propose to learn a model of the spatial, temporal and spectral characteristics of the nuisance, directly from the observations. In a pre-processing step, a statistical model of their correlations is built locally, and the data are centered and whitened to improve both their stationarity and signal-to-noise ratio (SNR). A convolutional neural network (CNN) is then trained in a supervised fashion to detect the residual signature of synthetic sources in the pre-processed images. Our method leads to a better trade-off between precision and recall than standard approaches in the field. It also outperforms a state-of-the-art algorithm based solely on a statistical framework. Besides, the exploitation of the spectral diversity improves the performance compared to a similar model built solely from spatio-temporal data.
Value at Risk (VaR) and Conditional Value at Risk (CVaR) have become the most popular measures of market risk in Financial and Insurance fields. However, the estimation of both risk measures is challenging, because it requires the knowledge of the tail of the distribution. Therefore, tools from Extreme Value Theory are usually employed, considering that the tail data follow a Generalized Pareto distribution (GPD). Using the existing relations from the parameters of the baseline distribution and the limit GPD's parameters, we define highly informative priors that incorporate all the information available for the whole set of observations. We show how to perform Metropolis-Hastings (MH) algorithm to estimate VaR and CVaR employing the highly informative priors, in the case of exponential, stable and Gamma distributions. Afterwards, we perform a thorough simulation study to compare the accuracy and precision provided by three different methods. Finally, data from a real example is analyzed to show the practical application of the methods.
Despite the proven significance of hyperspectral images (HSIs) in performing various computer vision tasks, its potential is adversely affected by the low-resolution (LR) property in the spatial domain, resulting from multiple physical factors. Inspired by recent advancements in deep generative models, we propose an HSI Super-resolution (SR) approach with Conditional Diffusion Models (HSR-Diff) that merges a high-resolution (HR) multispectral image (MSI) with the corresponding LR-HSI. HSR-Diff generates an HR-HSI via repeated refinement, in which the HR-HSI is initialized with pure Gaussian noise and iteratively refined. At each iteration, the noise is removed with a Conditional Denoising Transformer (CDF ormer) that is trained on denoising at different noise levels, conditioned on the hierarchical feature maps of HR-MSI and LR-HSI. In addition, a progressive learning strategy is employed to exploit the global information of full-resolution images. Systematic experiments have been conducted on four public datasets, demonstrating that HSR-Diff outperforms state-of-the-art methods.
Understanding the effect of a feature vector $x \in \mathbb{R}^d$ on the response value (label) $y \in \mathbb{R}$ is the cornerstone of many statistical learning problems. Ideally, it is desired to understand how a set of collected features combine together and influence the response value, but this problem is notoriously difficult, due to the high-dimensionality of data and limited number of labeled data points, among many others. In this work, we take a new perspective on this problem, and we study the question of assessing the difference of influence that the two given features have on the response value. We first propose a notion of closeness for the influence of features, and show that our definition recovers the familiar notion of the magnitude of coefficients in the parametric model. We then propose a novel method to test for the closeness of influence in general model-free supervised learning problems. Our proposed test can be used with finite number of samples with control on type I error rate, no matter the ground truth conditional law $\mathcal{L}(Y |X)$. We analyze the power of our test for two general learning problems i) linear regression, and ii) binary classification under mixture of Gaussian models, and show that under the proper choice of score function, an internal component of our test, with sufficient number of samples will achieve full statistical power. We evaluate our findings through extensive numerical simulations, specifically we adopt the datamodel framework (Ilyas, et al., 2022) for CIFAR-10 dataset to identify pairs of training samples with different influence on the trained model via optional black box training mechanisms.
In recent years, network models have gained prominence for their ability to capture complex associations. In statistical omics, networks can be used to model and study the functional relationships between genes, proteins, and other types of omics data. If a Gaussian graphical model is assumed, a gene association network can be determined from the non-zero entries of the inverse covariance matrix of the data. Due to the high-dimensional nature of such problems, integrative methods that leverage similarities between multiple graphical structures have become increasingly popular. The joint graphical lasso is a powerful tool for this purpose, however, the current AIC-based selection criterion used to tune the network sparsities and similarities leads to poor performance in high-dimensional settings. We propose stabJGL, which equips the joint graphical lasso with a stable and accurate penalty parameter selection approach that combines the notion of model stability with likelihood-based similarity selection. The resulting method makes the powerful joint graphical lasso available for use in omics settings, and outperforms the standard joint graphical lasso, as well as state-of-the-art joint methods, in terms of all performance measures we consider. Applying stabJGL to proteomic data from a pan-cancer study, we demonstrate the potential for novel discoveries the method brings. A user-friendly R package for stabJGL with tutorials is available on Github at //github.com/Camiling/stabJGL.
We establish globally optimal solutions to a class of fractional optimization problems on a class of constraint sets, whose key characteristics are as follows: 1) The numerator and the denominator of the objective function are both convex, semi-algebraic, Lipschitz continuous and differentiable with Lipschitz continuous gradients on the constraint set. 2) The constraint set is closed, convex and semi-algebraic. Compared with Dinkelbach's approach, our novelty falls into the following aspects: 1) Dinkelbach's has to solve a concave maximization problem in each iteration, which is nontrivial to obtain a solution, while ours only needs to conduct one proximity gradient operation in each iteration. 2) Dinkelbach's requires at least one nonnegative point for the numerator to proceed the algorithm, but ours does not, which is available to a much wider class of situations. 3) Dinkelbach's requires a closed and bounded constraint set, while ours only needs the closedness but not necessarily the boundedness. Therefore, our approach is viable for many more practical models, like optimizing the Sharpe ratio (SR) or the Information ratio in mathematical finance. Numerical experiments show that our approach achieves the ground-truth solutions in two simple examples. For real-world financial data, it outperforms several existing approaches for SR maximization.
We examine the problem of variance components testing in general mixed effects models using the likelihood ratio test. We account for the presence of nuisance parameters, i.e. the fact that some untested variances might also be equal to zero. Two main issues arise in this context leading to a non regular setting. First, under the null hypothesis the true parameter value lies on the boundary of the parameter space. Moreover, due to the presence of nuisance parameters the exact location of these boundary points is not known, which prevents from using classical asymptotic theory of maximum likelihood estimation. Then, in the specific context of nonlinear mixed-effects models, the Fisher information matrix is singular at the true parameter value. We address these two points by proposing a shrinked parametric bootstrap procedure, which is straightforward to apply even for nonlinear models. We show that the procedure is consistent, solving both the boundary and the singularity issues, and we provide a verifiable criterion for the applicability of our theoretical results. We show through a simulation study that, compared to the asymptotic approach, our procedure has a better small sample performance and is more robust to the presence of nuisance parameters. A real data application is also provided.
In this paper we describe a randomized algorithm which returns a maximal spanning forest of an unknown {\em weighted} undirected graph making $O(n)$ $\mathsf{CUT}$ queries in expectation. For weighted graphs, this is optimal due to a result in [Auza and Lee, 2021] which shows an $\Omega(n)$ lower bound for zero-error randomized algorithms. %To our knowledge, it is the only regime of this problem where we have upper and lower bounds tight up to constants. These questions have been extensively studied in the past few years, especially due to the problem's connections to symmetric submodular function minimization. We also describe a simple polynomial time deterministic algorithm that makes $O(\frac{n\log n}{\log\log n})$ queries on undirected unweighted graphs and returns a maximal spanning forest, thereby (slightly) improving upon the state-of-the-art.
Image segmentation is an important component of many image understanding systems. It aims to group pixels in a spatially and perceptually coherent manner. Typically, these algorithms have a collection of parameters that control the degree of over-segmentation produced. It still remains a challenge to properly select such parameters for human-like perceptual grouping. In this work, we exploit the diversity of segments produced by different choices of parameters. We scan the segmentation parameter space and generate a collection of image segmentation hypotheses (from highly over-segmented to under-segmented). These are fed into a cost minimization framework that produces the final segmentation by selecting segments that: (1) better describe the natural contours of the image, and (2) are more stable and persistent among all the segmentation hypotheses. We compare our algorithm's performance with state-of-the-art algorithms, showing that we can achieve improved results. We also show that our framework is robust to the choice of segmentation kernel that produces the initial set of hypotheses.
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