Fusing measurements from multiple, heterogeneous, partial sources, observing a common object or process, poses challenges due to the increasing availability of numbers and types of sensors. In this work we propose, implement and validate an end-to-end computational pipeline in the form of a multiple-auto-encoder neural network architecture for this task. The inputs to the pipeline are several sets of partial observations, and the result is a globally consistent latent space, harmonizing (rigidifying, fusing) all measurements. The key enabler is the availability of multiple slightly perturbed measurements of each instance:, local measurement, "bursts", that allows us to estimate the local distortion induced by each instrument. We demonstrate the approach in a sequence of examples, starting with simple two-dimensional data sets and proceeding to a Wi-Fi localization problem and to the solution of a "dynamical puzzle" arising in spatio-temporal observations of the solutions of Partial Differential Equations.
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In the realm of statistical exploration, the manipulation of pseudo-random values to discern their impact on data distribution presents a compelling avenue of inquiry. This article investigates the question: Is it possible to add pseudo-random values without compelling a shift towards a normal distribution?. Employing Python techniques, the study explores the nuances of pseudo-random value addition within the context of additions, aiming to unravel the interplay between randomness and resulting statistical characteristics. The Materials and Methods chapter details the construction of datasets comprising up to 300 billion pseudo-random values, employing three distinct layers of manipulation. The Results chapter visually and quantitatively explores the generated datasets, emphasizing distribution and standard deviation metrics. The study concludes with reflections on the implications of pseudo-random value manipulation and suggests avenues for future research. In the layered exploration, the first layer introduces subtle normalization with increasing summations, while the second layer enhances normality. The third layer disrupts typical distribution patterns, leaning towards randomness despite pseudo-random value summation. Standard deviation patterns across layers further illuminate the dynamic interplay of pseudo-random operations on statistical characteristics. While not aiming to disrupt academic norms, this work modestly contributes insights into data distribution complexities. Future studies are encouraged to delve deeper into the implications of data manipulation on statistical outcomes, extending the understanding of pseudo-random operations in diverse contexts.
The comparison of frequency distributions is a common statistical task with broad applications and a long history of methodological development. However, existing measures do not quantify the magnitude and direction by which one distribution is shifted relative to another. In the present study, we define distributional shift (DS) as the concentration of frequencies away from the greatest discrete class, e.g., a histogram's right-most bin. We derive a measure of DS based on the sum of cumulative frequencies, intuitively quantifying shift as a statistical moment. We then define relative distributional shift (RDS) as the difference in DS between distributions. Using simulated random sampling, we demonstrate that RDS is highly related to measures that are popularly used to compare frequency distributions. Focusing on a specific use case, i.e., simulated healthcare Evaluation and Management coding profiles, we show how RDS can be used to examine many pairs of empirical and expected distributions via shift-significance plots. In comparison to other measures, RDS has the unique advantage of being a signed (directional) measure based on a simple difference in an intuitive property.
This paper addresses the multiple two-sample test problem in a graph-structured setting, which is a common scenario in fields such as Spatial Statistics and Neuroscience. Each node $v$ in fixed graph deals with a two-sample testing problem between two node-specific probability density functions (pdfs), $p_v$ and $q_v$. The goal is to identify nodes where the null hypothesis $p_v = q_v$ should be rejected, under the assumption that connected nodes would yield similar test outcomes. We propose the non-parametric collaborative two-sample testing (CTST) framework that efficiently leverages the graph structure and minimizes the assumptions over $p_v$ and $q_v$. Our methodology integrates elements from f-divergence estimation, Kernel Methods, and Multitask Learning. We use synthetic experiments and a real sensor network detecting seismic activity to demonstrate that CTST outperforms state-of-the-art non-parametric statistical tests that apply at each node independently, hence disregard the geometry of the problem.
We introduce a method for computing immediately human interpretable yet accurate classifiers from tabular data. The classifiers obtained are short DNF-formulas, computed via first discretizing the original data to Boolean form and then using feature selection coupled with a very fast algorithm for producing the best possible Boolean classifier for the setting. We demonstrate the approach via 14 experiments, obtaining results with accuracies mainly similar to ones obtained via random forests, XGBoost, and existing results for the same datasets in the literature. In several cases, our approach in fact outperforms the reference results in relation to accuracy, even though the main objective of our study is the immediate interpretability of our classifiers. We also prove a new result on the probability that the classifier we obtain from real-life data corresponds to the ideally best classifier with respect to the background distribution the data comes from.
We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial "deflation" step to the standard generic chaining argument. The resulting tail bound is the sum of the complexity of the "deflated function class" in terms of a generalization of Talagrand's $\gamma$ functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cram\'{e}r functions. We also provide certain approximations for the mentioned seminorm when the function class lies in a given (exponential type) Orlicz space, that can be used to make the complexity term and the deviation term more explicit.
We study the problem of training diffusion models to sample from a distribution with a given unnormalized density or energy function. We benchmark several diffusion-structured inference methods, including simulation-based variational approaches and off-policy methods (continuous generative flow networks). Our results shed light on the relative advantages of existing algorithms while bringing into question some claims from past work. We also propose a novel exploration strategy for off-policy methods, based on local search in the target space with the use of a replay buffer, and show that it improves the quality of samples on a variety of target distributions. Our code for the sampling methods and benchmarks studied is made public at //github.com/GFNOrg/gfn-diffusion as a base for future work on diffusion models for amortized inference.
We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed approach to quantitative hypocoercivity based on space-time Poincar\'e inequalities can be rephrased and simplified in the language of lifts and how it can be applied to find optimal lifts.
We consider nonparametric Bayesian inference in a multidimensional diffusion model with reflecting boundary conditions based on discrete high-frequency observations. We prove a general posterior contraction rate theorem in $L^2$-loss, which is applied to Gaussian priors. The resulting posteriors, as well as their posterior means, are shown to converge to the ground truth at the minimax optimal rate over H\"older smoothness classes in any dimension. Of independent interest and as part of our proofs, we show that certain frequentist penalized least squares estimators are also minimax optimal.
Collecting large quantities of high-quality data is often prohibitively expensive or impractical, and a crucial bottleneck in machine learning. One may instead augment a small set of $n$ data points from the target distribution with data from more accessible sources like public datasets, data collected under different circumstances, or synthesized by generative models. Blurring distinctions, we refer to such data as `surrogate data'. We define a simple scheme for integrating surrogate data into training and use both theoretical models and empirical studies to explore its behavior. Our main findings are: $(i)$ Integrating surrogate data can significantly reduce the test error on the original distribution; $(ii)$ In order to reap this benefit, it is crucial to use optimally weighted empirical risk minimization; $(iii)$ The test error of models trained on mixtures of real and surrogate data is well described by a scaling law. This can be used to predict the optimal weighting and the gain from surrogate data.
An emerging trend in approximate counting is to show that certain `low-temperature' problems are easy on typical instances, despite worst-case hardness results. For the class of regular graphs one usually shows that expansion can be exploited algorithmically, and since random regular graphs are good expanders with high probability the problem is typically tractable. Inspired by approaches used in subexponential-time algorithms for Unique Games, we develop an approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition. In such graphs it may not suffice to explore the state space of the model close to ground states, and a novel feature of our method is to efficiently find a larger set of `pseudo-ground states' such that it is enough to explore the model around each pseudo-ground state.
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