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Bruno Jedynak

Publications and source records attributed to Bruno Jedynak.

10 recordsLinked to original sources
arXiv2025

The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations

We present a novel kernel-based method for learning multivariate stochastic differential equations (SDEs). The method follows a two-step procedure: we first estimate the drift term function, then the (matrix-valued) diffusion function given the drift. Occupation kernels are integral functionals on a reproducing kernel Hilbert space (RKHS) that aggregate information over a trajectory. Our approach leverages vector-valued occupation kernels for estimating the drift component of the stochastic process. For diffusion estimation, we extend this framework by introducing operator-valued occupation kernels, enabling the estimation of an auxiliary matrix-valued function as a positive semi-definite operator, from which we readily derive the diffusion estimate. This enables us to avoid common challenges in SDE learning, such as intractable likelihoods, by optimizing a reconstruction-error-based objective. We propose a simple learning procedure that retains strong predictive accuracy while using Fenchel duality to promote efficiency. We validate the method on simulated benchmarks and a real-world dataset of Amyloid imaging in healthy and Alzheimer's disease subjects.

stat.ML↗
arXiv2025

ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spaces

We present a Representer Theorem result for a large class of weak formulation problems. We provide examples of applications of our formulation both in traditional machine learning and numerical methods as well as in new and emerging techniques. Finally we apply our formulation to generalize the multivariate occupation kernel (MOCK) method for learning dynamical systems from data proposing the more general Riesz Occupation Kernel (ROCK) method. Our generalized methods are both more computationally efficient and performant on most of the benchmarks we test against.

stat.ML↗
arXiv2024

The Stochastic Occupation Kernel Method for System Identification

The method of occupation kernels has been used to learn ordinary differential equations from data in a non-parametric way. We propose a two-step method for learning the drift and diffusion of a stochastic differential equation given snapshots of the process. In the first step, we learn the drift by applying the occupation kernel algorithm to the expected value of the process. In the second step, we learn the diffusion given the drift using a semi-definite program. Specifically, we learn the diffusion squared as a non-negative function in a RKHS associated with the square of a kernel. We present examples and simulations.

stat.ML↗
arXiv2023

A Finite-Horizon Approach to Active Level Set Estimation

We consider the problem of active learning in the context of spatial sampling for level set estimation (LSE), where the goal is to localize all regions where a function of interest lies above/below a given threshold as quickly as possible. We present a finite-horizon search procedure to perform LSE in one dimension while optimally balancing both the final estimation error and the distance traveled for a fixed number of samples. A tuning parameter is used to trade off between the estimation accuracy and distance traveled. We show that the resulting optimization problem can be solved in closed form and that the resulting policy generalizes existing approaches to this problem. We then show how this approach can be used to perform level set estimation in higher dimensions under the popular Gaussian process model. Empirical results on synthetic data indicate that as the cost of travel increases, our method's ability to treat distance nonmyopically allows it to significantly improve on the state of the art. On real air quality data, our approach achieves roughly one fifth the estimation error at less than half the cost of competing algorithms.

cs.LG↗
arXiv2023

MOCK: an Algorithm for Learning Nonparametric Differential Equations via Multivariate Occupation Kernel Functions

Learning a nonparametric system of ordinary differential equations from trajectories in a $d$-dimensional state space requires learning $d$ functions of $d$ variables. Explicit formulations often scale quadratically in $d$ unless additional knowledge about system properties, such as sparsity and symmetries, is available. In this work, we propose a linear approach, the multivariate occupation kernel method (MOCK), using the implicit formulation provided by vector-valued reproducing kernel Hilbert spaces. The solution for the vector field relies on multivariate occupation kernel functions associated with the trajectories and scales linearly with the dimension of the state space. We validate through experiments on a variety of simulated and real datasets ranging from 2 to 1024 dimensions. MOCK outperforms all other comparators on 3 of the 9 datasets on full trajectory prediction and 4 out of the 9 datasets on next-point prediction.

stat.ML↗
arXiv2021

The Alzheimer's Disease Prediction Of Longitudinal Evolution (TADPOLE) Challenge: Results after 1 Year Follow-up

We present the findings of "The Alzheimer's Disease Prediction Of Longitudinal Evolution" (TADPOLE) Challenge, which compared the performance of 92 algorithms from 33 international teams at predicting the future trajectory of 219 individuals at risk of Alzheimer's disease. Challenge participants were required to make a prediction, for each month of a 5-year future time period, of three key outcomes: clinical diagnosis, Alzheimer's Disease Assessment Scale Cognitive Subdomain (ADAS-Cog13), and total volume of the ventricles. The methods used by challenge participants included multivariate linear regression, machine learning methods such as support vector machines and deep neural networks, as well as disease progression models. No single submission was best at predicting all three outcomes. For clinical diagnosis and ventricle volume prediction, the best algorithms strongly outperform simple baselines in predictive ability. However, for ADAS-Cog13 no single submitted prediction method was significantly better than random guesswork. Two ensemble methods based on taking the mean and median over all predictions, obtained top scores on almost all tasks. Better than average performance at diagnosis prediction was generally associated with the additional inclusion of features from cerebrospinal fluid (CSF) samples and diffusion tensor imaging (DTI). On the other hand, better performance at ventricle volume prediction was associated with inclusion of summary statistics, such as the slope or maxima/minima of biomarkers. TADPOLE's unique results suggest that current prediction algorithms provide sufficient accuracy to exploit biomarkers related to clinical diagnosis and ventricle volume, for cohort refinement in clinical trials for Alzheimer's disease. However, results call into question the usage of cognitive test scores for patient selection and as a primary endpoint in clinical trials.

q-bio.PE↗
arXiv2017

Gaussian Processes with Context-Supported Priors for Active Object Localization

We devise an algorithm using a Bayesian optimization framework in conjunction with contextual visual data for the efficient localization of objects in still images. Recent research has demonstrated substantial progress in object localization and related tasks for computer vision. However, many current state-of-the-art object localization procedures still suffer from inaccuracy and inefficiency, in addition to failing to provide a principled and interpretable system amenable to high-level vision tasks. We address these issues with the current research. Our method encompasses an active search procedure that uses contextual data to generate initial bounding-box proposals for a target object. We train a convolutional neural network to approximate an offset distance from the target object. Next, we use a Gaussian Process to model this offset response signal over the search space of the target. We then employ a Bayesian active search for accurate localization of the target. In experiments, we compare our approach to a state-of-theart bounding-box regression method for a challenging pedestrian localization task. Our method exhibits a substantial improvement over this baseline regression method.

cs.CV↗
arXiv2015

Non-Adaptive Policies for 20 Questions Target Localization

The problem of target localization with noise is addressed. The target is a sample from a continuous random variable with known distribution and the goal is to locate it with minimum mean squared error distortion. The localization scheme or policy proceeds by queries, or questions, weather or not the target belongs to some subset as it is addressed in the 20-question framework. These subsets are not constrained to be intervals and the answers to the queries are noisy. While this situation is well studied for adaptive querying, this paper is focused on the non adaptive querying policies based on dyadic questions. The asymptotic minimum achievable distortion under such policies is derived. Furthermore, a policy named the Aurelian1 is exhibited which achieves asymptotically this distortion.

cs.IT↗
arXiv2010

Active Testing for Face Detection and Localization

We provide a novel search technique, which uses a hierarchical model and a mutual information gain heuristic to efficiently prune the search space when localizing faces in images. We show exponential gains in computation over traditional sliding window approaches, while keeping similar performance levels.

cs.CV↗