Random projections constitute a simple, yet effective technique for dimensionality reduction with applications in learning and search problems. In the present paper, we consider the problem of estimating cosine similarities when the projected data undergo scalar quantization to b bits. We here argue that the maximum likelihood estimator (MLE) is a principled approach to deal with the non-linearity resulting from quantization, and subsequently study its computational and statistical properties. A specific focus is on the on the trade-off between bit depth and the number of projections given a fixed budget of bits for storage or transmission. Along the way, we also touch upon the existence of a qualitative counterpart to the Johnson-Lindenstrauss lemma in the presence of quantization.

Accurately differentiating between what are truly unpredictably random and systematic changes that occur at random can have profound effect on affect and cognition. To examine the underlying computational principles that guide different learning behavior in an uncertain environment, we compared an R-W model and a Bayesian approach in a visual search task with different volatility levels. Both R-W model and the Bayesian approach reflected an individual's estimation of the environmental volatility, and there is a strong correlation between the learning rate in R-W model and the belief of stationarity in the Bayesian approach in different volatility conditions. In a low volatility condition, R-W model indicates that learning rate positively correlates with lose-shift rate, but not choice optimality (inverted U shape). The Bayesian approach indicates that the belief of environmental stationarity positively correlates with choice optimality, but not lose-shift rate (inverted U shape). In addition, we showed that comparing to Expert learners, individuals with high lose-shift rate (sub-optimal learners) had significantly higher learning rate estimated from R-W model and lower belief of stationarity from the Bayesian model.

We propose probabilistic latent variable models for multi-view anomaly detection, which is the task of finding instances that have inconsistent views given multi-view data. With the proposed model, all views of a non-anomalous instance are assumed to be generated from a single latent vector. On the other hand, an anomalous instance is assumed to have multiple latent vectors, and its different views are generated from different latent vectors. By inferring the number of latent vectors used for each instance with Dirichlet process priors, we obtain multiview anomaly scores. The proposed model can be seen as a robust extension of probabilistic canonical correlation analysis for noisy multi-view data. We present Bayesian inference procedures for the proposed model based on a stochastic EM algorithm. The effectiveness of the proposed model is demonstrated in terms of performance when detecting multi-view anomalies.

Bayes type methods, which would otherwise be well suited to the i.i.d. Our focus in this paper is not to explore the generalities of finite-dimensional processes.

Estimating patient's clinical state from multiple concurrent physiological streams plays an important role in determining if a therapeutic intervention is necessary and for triaging patients in the hospital. In this paper we construct a non-parametric learning algorithm to estimate the clinical state of a patient. The algorithm addresses several known challenges with clinical state estimation such as eliminating the bias introduced by therapeutic intervention censoring, increasing the timeliness of state estimation while ensuring a sufficient accuracy, and the ability to detect anomalous clinical states. These benefits are obtained by combining the tools of non-parametric Bayesian inference, permutation testing, and generalizations of the empirical Bernstein inequality. The algorithm is validated using real-world data from a cancer ward in a large academic hospital.

We propose an operator learning approach to accelerate geometric Markov chain Monte Carlo (MCMC) for solving infinite-dimensional nonlinear Bayesian inverse problems. While geometric MCMC employs high-quality proposals that adapt to posterior local geometry, it requires computing local gradient and Hessian information of the log-likelihood, incurring a high cost when the parameter-to-observable (PtO) map is defined through expensive model simulations. We consider a delayed-acceptance geometric MCMC method driven by a neural operator surrogate of the PtO map, where the proposal is designed to exploit fast surrogate approximations of the log-likelihood and, simultaneously, its gradient and Hessian. To achieve a substantial speedup, the surrogate needs to be accurate in predicting both the observable and its parametric derivative (the derivative of the observable with respect to the parameter). Training such a surrogate via conventional operator learning using input--output samples often demands a prohibitively large number of model simulations. In this work, we present an extension of derivative-informed operator learning [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] using input--output--derivative training samples. Such a learning method leads to derivative-informed neural operator (DINO) surrogates that accurately predict the observable and its parametric derivative at a significantly lower training cost than the conventional method. Cost and error analysis for reduced basis DINO surrogates are provided. Numerical studies on PDE-constrained Bayesian inversion demonstrate that DINO-driven MCMC generates effective posterior samples 3--9 times faster than geometric MCMC and 60--97 times faster than prior geometry-based MCMC. Furthermore, the training cost of DINO surrogates breaks even after collecting merely 10--25 effective posterior samples compared to geometric MCMC.

In the context of developing nations like India, traditional business to business (B2B) commerce heavily relies on the establishment of robust relationships, trust, and credit arrangements between buyers and sellers. Consequently, ecommerce enterprises frequently. Established in 2016 with a vision to revolutionize trade in India through technology, Udaan is the countrys largest business to business ecommerce platform. Udaan operates across diverse product categories, including lifestyle, electronics, home and employ telecallers to cultivate buyer relationships, streamline order placement procedures, and promote special promotions. The accurate anticipation of buyer order placement behavior emerges as a pivotal factor for attaining sustainable growth, heightening competitiveness, and optimizing the efficiency of these telecallers. To address this challenge, we have employed an ensemble approach comprising XGBoost and a modified version of Poisson Gamma model to predict customer order patterns with precision. This paper provides an in-depth exploration of the strategic fusion of machine learning and an empirical Bayesian approach, bolstered by the judicious selection of pertinent features. This innovative approach has yielded a remarkable 3 times increase in customer order rates, show casing its potential for transformative impact in the ecommerce industry.

Spatiotemporal datasets, which consist of spatially-referenced time series, are ubiquitous in many scientific and business-intelligence applications, such as air pollution monitoring, disease tracking, and cloud-demand forecasting. As modern datasets continue to increase in size and complexity, there is a growing need for new statistical methods that are flexible enough to capture complex spatiotemporal dynamics and scalable enough to handle large prediction problems. This work presents the Bayesian Neural Field (BayesNF), a domain-general statistical model for inferring rich probability distributions over a spatiotemporal domain, which can be used for data-analysis tasks including forecasting, interpolation, and variography. BayesNF integrates a novel deep neural network architecture for high-capacity function estimation with hierarchical Bayesian inference for robust uncertainty quantification. By defining the prior through a sequence of smooth differentiable transforms, posterior inference is conducted on large-scale data using variationally learned surrogates trained via stochastic gradient descent. We evaluate BayesNF against prominent statistical and machine-learning baselines, showing considerable improvements on diverse prediction problems from climate and public health datasets that contain tens to hundreds of thousands of measurements. The paper is accompanied with an open-source software package (https://github.com/google/bayesnf) that is easy-to-use and compatible with modern GPU and TPU accelerators on the JAX machine learning platform.

We study when low coordinate degree functions (LCDF) -- linear combinations of functions depending on small subsets of entries of a vector -- can hypothesis test between high-dimensional probability measures. These functions are a generalization, proposed in Hopkins' 2018 thesis but seldom studied since, of low degree polynomials (LDP), a class widely used in recent literature as a proxy for all efficient algorithms for tasks in statistics and optimization. Instead of the orthogonal polynomial decompositions used in LDP calculations, our analysis of LCDF is based on the Efron-Stein or ANOVA decomposition, making it much more broadly applicable. By way of illustration, we prove channel universality for the success of LCDF in testing for the presence of sufficiently "dilute" random signals through noisy channels: the efficacy of LCDF depends on the channel only through the scalar Fisher information for a class of channels including nearly arbitrary additive i.i.d. noise and nearly arbitrary exponential families. As applications, we extend lower bounds against LDP for spiked matrix and tensor models under additive Gaussian noise to lower bounds against LCDF under general noisy channels. We also give a simple and unified treatment of the effect of censoring models by erasing observations at random and of quantizing models by taking the sign of the observations. These results are the first computational lower bounds against any large class of algorithms for all of these models when the channel is not one of a few special cases, and thereby give the first substantial evidence for the universality of several statistical-to-computational gaps.

We show that several striking differences in memory performance between recognition and recall tasks are explained by an ecological bias endemic in classic memory experiments - that such experiments universally involve more stimuli than retrieval cues. We show that while it is sensible to think of recall as simply retrieving items when probed with a cue - typically the item list itself - it is better to think of recognition as retrieving cues when probed with items. To test this theory, by manipulating the number of items and cues in a memory experiment, we show a crossover effect in memory performance within subjects such that recognition performance is superior to recall performance when the number of items is greater than the number of cues and recall performance is better than recognition when the converse holds. We build a simple computational model around this theory, using sampling to approximate an ideal Bayesian observer encoding and retrieving situational co-occurrence frequencies of stimuli and retrieval cues. This model robustly reproduces a number of dissociations in recognition and recall previously used to argue for dual-process accounts of declarative memory.