arXiv subjects

Houman Owhadi

Publications and source records attributed to Houman Owhadi.

At least 55 records · Page 3Linked to original sources

arXiv2024

Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation

The article presents a systematic study of the problem of conditioning a Gaussian random variable $ξ$ on nonlinear observations of the form $F \circ ϕ(ξ)$ where $ϕ: \mathcal{X} \to \mathbb{R}^N$ is a bounded linear operator and $F$ is nonlinear. Such problems arise in the context of Bayesian inference and recent machine learning-inspired PDE solvers. We give a representer theorem for the conditioned random variable $ξ\mid F\circ ϕ(ξ)$, stating that it decomposes as the sum of an infinite-dimensional Gaussian (which is identified analytically) as well as a finite-dimensional non-Gaussian measure. We also introduce a novel notion of the mode of a conditional measure by taking the limit of the natural relaxation of the problem, to which we can apply the existing notion of maximum a posteriori estimators of posterior measures. Finally, we introduce a variant of the Laplace approximation for the efficient simulation of the aforementioned conditioned Gaussian random variables towards uncertainty quantification.

stat.ML↗

arXiv2024

Optimization-based frequentist confidence intervals for functionals in constrained inverse problems: Resolving the Burrus conjecture

We present an optimization-based framework to construct confidence intervals for functionals in constrained inverse problems, ensuring valid one-at-a-time frequentist coverage guarantees. Our approach builds upon the now-called strict bounds intervals, originally pioneered by Burrus (1965) and Rust and Burrus (1972), which offer ways to directly incorporate any side information about the parameters during inference without introducing external biases. This family of methods allows for uncertainty quantification in ill-posed inverse problems without needing to select a regularizing prior. By tying optimization-based intervals to an inversion of a constrained likelihood ratio test, we translate interval coverage guarantees into type I error control and characterize the resulting interval via solutions to optimization problems. Along the way, we refute the Burrus conjecture, which posited that, for possibly rank-deficient linear Gaussian models with positivity constraints, a correction based on the quantile of the chi-squared distribution with one degree of freedom suffices to shorten intervals while maintaining frequentist coverage guarantees. Our framework provides a novel approach to analyzing the conjecture, and we construct a counterexample employing a stochastic dominance argument, which we also use to disprove a general form of the conjecture. We illustrate our framework with several numerical examples and provide directions for extensions beyond the Rust-Burrus method for nonlinear, non-Gaussian settings with general constraints.

math.ST↗

arXiv2024

Bridging Algorithmic Information Theory and Machine Learning: A New Approach to Kernel Learning

Machine Learning (ML) and Algorithmic Information Theory (AIT) look at Complexity from different points of view. We explore the interface between AIT and Kernel Methods (that are prevalent in ML) by adopting an AIT perspective on the problem of learning kernels from data, in kernel ridge regression, through the method of Sparse Kernel Flows. In particular, by looking at the differences and commonalities between Minimal Description Length (MDL) and Regularization in Machine Learning (RML), we prove that the method of Sparse Kernel Flows is the natural approach to adopt to learn kernels from data. This approach aligns naturally with the MDL principle, offering a more robust theoretical basis than the existing reliance on cross-validation. The study reveals that deriving Sparse Kernel Flows does not require a statistical approach; instead, one can directly engage with code-lengths and complexities, concepts central to AIT. Thereby, this approach opens the door to reformulating algorithms in machine learning using tools from AIT, with the aim of providing them a more solid theoretical foundation.

cs.LG↗

arXiv2024

Can Specific THz Fields Induce Collective Base-Flipping in DNA? A Stochastic Averaging and Resonant Enhancement Investigation Based on a New Mesoscopic Model

We study the metastability, internal frequencies, activation mechanism, energy transfer, and the collective base-flipping in a mesoscopic DNA via resonance with specific electric fields. Our new mesoscopic DNA model takes into account not only the issues of helicity and the coupling of an electric field with the base dipole moments, but also includes environmental effects such as fluid viscosity and thermal noise. And all the parameter values are chosen to best represent the typical values for the opening and closing dynamics of a DNA. Our study shows that while the mesocopic DNA is metastable and robust to environmental effects, it is vulnerable to certain frequencies that could be targeted by specific THz fields for triggering its collective base-flipping dynamics and causing large amplitude separation of base pairs. Based on applying Freidlin-Wentzell method of stochastic averaging and the newly developed theory of resonant enhancement to our mesoscopic DNA model, our semi-analytic estimates show that the required fields should be THz fields with frequencies around 0.28 THz and with amplitudes in the order of 450 kV/cm. These estimates compare well with the experimental data of Titova et al., which have demonstrated that they could affect the function of DNA in human skin tissues by THz pulses with frequencies around 0.5 THz and with a peak electric field at 220 kV/cm. Moreover, our estimates also conform to a number of other experimental results which appeared in the last couple years.

physics.bio-ph↗

arXiv2024

Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its \textit{partial derivatives}, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute $ε$-approximate inverse Cholesky factors of the kernel matrices with complexity $O(N\log^d(N/ε))$ in space and $O(N\log^{2d}(N/ε))$ in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm's near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations.

math.NA↗

arXiv2024

Diffeomorphic Measure Matching with Kernels for Generative Modeling

This article presents a general framework for the transport of probability measures towards minimum divergence generative modeling and sampling using ordinary differential equations (ODEs) and Reproducing Kernel Hilbert Spaces (RKHSs), inspired by ideas from diffeomorphic matching and image registration. A theoretical analysis of the proposed method is presented, giving a priori error bounds in terms of the complexity of the model, the number of samples in the training set, and model misspecification. An extensive suite of numerical experiments further highlights the properties, strengths, and weaknesses of the method and extends its applicability to other tasks, such as conditional simulation and inference.

stat.ML↗

arXiv2024

A Mini-Batch Method for Solving Nonlinear PDEs with Gaussian Processes

Gaussian processes (GPs) based methods for solving partial differential equations (PDEs) demonstrate great promise by bridging the gap between the theoretical rigor of traditional numerical algorithms and the flexible design of machine learning solvers. The main bottleneck of GP methods lies in the inversion of a covariance matrix, whose cost grows cubically concerning the size of samples. Drawing inspiration from neural networks, we propose a mini-batch algorithm combined with GPs to solve nonlinear PDEs. A naive deployment of a stochastic gradient descent method for solving PDEs with GPs is challenging, as the objective function in the requisite minimization problem cannot be depicted as the expectation of a finite-dimensional random function. To address this issue, we employ a mini-batch method to the corresponding infinite-dimensional minimization problem over function spaces. The algorithm takes a mini-batch of samples at each step to update the GP model. Thus, the computational cost is allotted to each iteration. Using stability analysis and convexity arguments, we show that the mini-batch method steadily reduces a natural measure of errors towards zero at the rate of $O(1/K+1/M)$, where $K$ is the number of iterations and $M$ is the batch size.

math.NA↗

arXiv2023

Kernel Methods are Competitive for Operator Learning

We present a general kernel-based framework for learning operators between Banach spaces along with a priori error analysis and comprehensive numerical comparisons with popular neural net (NN) approaches such as Deep Operator Net (DeepONet) [Lu et al.] and Fourier Neural Operator (FNO) [Li et al.]. We consider the setting where the input/output spaces of target operator $\mathcal{G}^\dagger\,:\, \mathcal{U}\to \mathcal{V}$ are reproducing kernel Hilbert spaces (RKHS), the data comes in the form of partial observations $ϕ(u_i), φ(v_i)$ of input/output functions $v_i=\mathcal{G}^\dagger(u_i)$ ($i=1,\ldots,N$), and the measurement operators $ϕ\,:\, \mathcal{U}\to \mathbb{R}^n$ and $φ\,:\, \mathcal{V} \to \mathbb{R}^m$ are linear. Writing $ψ\,:\, \mathbb{R}^n \to \mathcal{U}$ and $χ\,:\, \mathbb{R}^m \to \mathcal{V}$ for the optimal recovery maps associated with $ϕ$ and $φ$, we approximate $\mathcal{G}^\dagger$ with $\bar{\mathcal{G}}=χ\circ \bar{f} \circ ϕ$ where $\bar{f}$ is an optimal recovery approximation of $f^\dagger:=φ\circ \mathcal{G}^\dagger \circ ψ\,:\,\mathbb{R}^n \to \mathbb{R}^m$. We show that, even when using vanilla kernels (e.g., linear or Matérn), our approach is competitive in terms of cost-accuracy trade-off and either matches or beats the performance of NN methods on a majority of benchmarks. Additionally, our framework offers several advantages inherited from kernel methods: simplicity, interpretability, convergence guarantees, a priori error estimates, and Bayesian uncertainty quantification. As such, it can serve as a natural benchmark for operator learning.

stat.ML↗

arXiv2023

Sparse recovery of elliptic solvers from matrix-vector products

In this work, we show that solvers of elliptic boundary value problems in $d$ dimensions can be approximated to accuracy $ε$ from only $\mathcal{O}\left(\log(N)\log^{d}(N / ε)\right)$ matrix-vector products with carefully chosen vectors (right-hand sides). The solver is only accessed as a black box, and the underlying operator may be unknown and of an arbitrarily high order. Our algorithm (1) has complexity $\mathcal{O}\left(N\log^2(N)\log^{2d}(N / ε)\right)$ and represents the solution operator as a sparse Cholesky factorization with $\mathcal{O}\left(N\log(N)\log^{d}(N / ε)\right)$ nonzero entries, (2) allows for embarrassingly parallel evaluation of the solution operator and the computation of its log-determinant, (3) allows for $\mathcal{O}\left(\log(N)\log^{d}(N / ε)\right)$ complexity computation of individual entries of the matrix representation of the solver that, in turn, enables its recompression to an $\mathcal{O}\left(N\log^{d}(N / ε)\right)$ complexity representation. As a byproduct, our compression scheme produces a homogenized solution operator with near-optimal approximation accuracy. By polynomial approximation, we can also approximate the continuous Green's function (in operator and Hilbert-Schmidt norm) to accuracy $ε$ from $\mathcal{O}\left(\log^{1 + d}\left(ε^{-1}\right)\right)$ solutions of the PDE. We include rigorous proofs of these results. To the best of our knowledge, our algorithm achieves the best known trade-off between accuracy $ε$ and the number of required matrix-vector products.

math.NA↗

arXiv2023

Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

math.NA↗

arXiv2023

Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems

Regressing the vector field of a dynamical system from a finite number of observed states is a natural way to learn surrogate models for such systems. A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a data-adapted kernel which can be learned by using Kernel Flows. The method of Kernel Flows is a trainable machine learning method that learns the optimal parameters of a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used. The objective function could be a short-term prediction or some other objective for other variants of Kernel Flows). However, this method is limited by the choice of the base kernel. In this paper, we introduce the method of \emph{Sparse Kernel Flows } in order to learn the ``best'' kernel by starting from a large dictionary of kernels. It is based on sparsifying a kernel that is a linear combination of elemental kernels. We apply this approach to a library of 132 chaotic systems.

stat.ML↗

arXiv2023

Gaussian Process Hydrodynamics

We present a Gaussian Process (GP) approach (Gaussian Process Hydrodynamics, GPH) for approximating the solution of the Euler and Navier-Stokes equations. As in Smoothed Particle Hydrodynamics (SPH), GPH is a Lagrangian particle-based approach involving the tracking of a finite number of particles transported by the flow. However, these particles do not represent mollified particles of matter but carry discrete/partial information about the continuous flow. Closure is achieved by placing a divergence-free GP prior $ξ$ on the velocity field and conditioning on vorticity at particle locations. Known physics (e.g., the Richardson cascade and velocity-increments power laws) is incorporated into the GP prior through physics-informed additive kernels. This approach allows us to coarse-grain turbulence in a statistical manner rather than a deterministic one. By enforcing incompressibility and fluid/structure boundary conditions through the selection of the kernel, GPH requires much fewer particles than SPH. Since GPH has a natural probabilistic interpretation, numerical results come with uncertainty estimates enabling their incorporation into a UQ pipeline and the adding/removing of particles (quantas of information) in an adapted manner. The proposed approach is amenable to analysis, it inherits the complexity of state-of-the-art solvers for dense kernel matrices, and it leads to a natural definition of turbulence as information loss. Numerical experiments support the importance of selecting physics-informed kernels and illustrate the major impact of such kernels on accuracy and stability. Since the proposed approach has a Bayesian interpretation, it naturally enables data assimilation and making predictions and estimations based on mixing simulation data with experimental data.

physics.flu-dyn↗

arXiv2022

Multiclass classification utilising an estimated algorithmic probability prior

Methods of pattern recognition and machine learning are applied extensively in science, technology, and society. Hence, any advances in related theory may translate into large-scale impact. Here we explore how algorithmic information theory, especially algorithmic probability, may aid in a machine learning task. We study a multiclass supervised classification problem, namely learning the RNA molecule sequence-to-shape map, where the different possible shapes are taken to be the classes. The primary motivation for this work is a proof of concept example, where a concrete, well-motivated machine learning task can be aided by approximations to algorithmic probability. Our approach is based on directly estimating the class (i.e., shape) probabilities from shape complexities, and using the estimated probabilities as a prior in a Gaussian process learning problem. Naturally, with a large amount of training data, the prior has no significant influence on classification accuracy, but in the very small training data regime, we show that using the prior can substantially improve classification accuracy. To our knowledge, this work is one of the first to demonstrate how algorithmic probability can aid in a concrete, real-world, machine learning problem.

cs.LG↗

arXiv2022

One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels

We consider the problem of learning Stochastic Differential Equations of the form $dX_t = f(X_t)dt+σ(X_t)dW_t $ from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions $f$, $σ$, and stochastic process $dW_t$ representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion and data adapted kernels learned via a new variant of cross validation. Our approach can be decomposed as follows: (1) Represent the time-increment map $X_t \rightarrow X_{t+dt}$ as a Computational Graph in which $f$, $σ$ and $dW_t$ appear as unknown functions and random variables. (2) Complete the graph (approximate unknown functions and random variables) via Maximum a Posteriori Estimation (given the data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions (kernels) of the GP priors from data with randomized cross-validation. Numerical experiments illustrate the efficacy, robustness, and scope of our method.

stat.ML↗

arXiv2022

Do ideas have shape? Idea registration as the continuous limit of artificial neural networks

We introduce a GP generalization of ResNets (including ResNets as a particular case). We show that ResNets (and their GP generalization) converge, in the infinite depth limit, to a generalization of image registration variational algorithms. Whereas computational anatomy aligns images via warping of the material space, this generalization aligns ideas (or abstract shapes as in Plato's theory of forms) via the warping of the RKHS of functions mapping the input space to the output space. While the Hamiltonian interpretation of ResNets is not new, it was based on an Ansatz. We do not rely on this Ansatz and present the first rigorous proof of convergence of ResNets with trained weights and biases towards a Hamiltonian dynamics driven flow. Our constructive proof reveals several remarkable properties of ResNets and their GP generalization. ResNets regressors are kernel regressors with data-dependent warping kernels. Minimizers of $L_2$ regularized ResNets satisfy a discrete least action principle implying the near preservation of the norm of weights and biases across layers. The trained weights of ResNets with $L^2$ regularization can be identified by solving an autonomous Hamiltonian system. The trained ResNet parameters are unique up to the initial momentum whose representation is generally sparse. The kernel regularization strategy provides a provably robust alternative to Dropout for ANNs. We introduce a functional generalization of GPs leading to error estimates for ResNets. We identify the (EPDiff) mean fields limit of trained ResNet parameters. We show that the composition of warping regression blocks with reduced equivariant multichannel kernels (introduced here) recovers and generalizes CNNs to arbitrary spaces and groups of transformations.

stat.ML↗

arXiv2022

Uncertainty Quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball

There are essentially three kinds of approaches to Uncertainty Quantification (UQ): (A) robust optimization, (B) Bayesian, (C) decision theory. Although (A) is robust, it is unfavorable with respect to accuracy and data assimilation. (B) requires a prior, it is generally brittle and posterior estimations can be slow. Although (C) leads to the identification of an optimal prior, its approximation suffers from the curse of dimensionality and the notion of risk is one that is averaged with respect to the distribution of the data. We introduce a 4th kind which is a hybrid between (A), (B), (C), and hypothesis testing. It can be summarized as, after observing a sample $x$, (1) defining a likelihood region through the relative likelihood and (2) playing a minmax game in that region to define optimal estimators and their risk. The resulting method has several desirable properties (a) an optimal prior is identified after measuring the data, and the notion of risk is a posterior one, (b) the determination of the optimal estimate and its risk can be reduced to computing the minimum enclosing ball of the image of the likelihood region under the quantity of interest map (which is fast and not subject to the curse of dimensionality). The method is characterized by a parameter in $ [0,1]$ acting as an assumed lower bound on the rarity of the observed data (the relative likelihood). When that parameter is near $1$, the method produces a posterior distribution concentrated around a maximum likelihood estimate with tight but low confidence UQ estimates. When that parameter is near $0$, the method produces a maximal risk posterior distribution with high confidence UQ estimates. In addition to navigating the accuracy-uncertainty tradeoff, the proposed method addresses the brittleness of Bayesian inference by navigating the robustness-accuracy tradeoff associated with data assimilation.

stat.ME↗

arXiv2022

Learning "best" kernels from data in Gaussian process regression. With application to aerodynamics

This paper introduces algorithms to select/design kernels in Gaussian process regression/kriging surrogate modeling techniques. We adopt the setting of kernel method solutions in ad hoc functional spaces, namely Reproducing Kernel Hilbert Spaces (RKHS), to solve the problem of approximating a regular target function given observations of it, i.e. supervised learning. A first class of algorithms is kernel flow, which was introduced in the context of classification in machine learning. It can be seen as a cross-validation procedure whereby a "best" kernel is selected such that the loss of accuracy incurred by removing some part of the dataset (typically half of it) is minimized. A second class of algorithms is called spectral kernel ridge regression, and aims at selecting a "best" kernel such that the norm of the function to be approximated is minimal in the associated RKHS. Within Mercer's theorem framework, we obtain an explicit construction of that "best" kernel in terms of the main features of the target function. Both approaches of learning kernels from data are illustrated by numerical examples on synthetic test functions, and on a classical test case in turbulence modeling validation for transonic flows about a two-dimensional airfoil.

stat.ML↗

arXiv2022

Computational Graph Completion

We introduce a framework for generating, organizing, and reasoning with computational knowledge. It is motivated by the observation that most problems in Computational Sciences and Engineering (CSE) can be formulated as that of completing (from data) a computational graph (or hypergraph) representing dependencies between functions and variables. Nodes represent variables, and edges represent functions. Functions and variables may be known, unknown, or random. Data comes in the form of observations of distinct values of a finite number of subsets of the variables of the graph (satisfying its functional dependencies). The underlying problem combines a regression problem (approximating unknown functions) with a matrix completion problem (recovering unobserved variables in the data). Replacing unknown functions by Gaussian Processes (GPs) and conditioning on observed data provides a simple but efficient approach to completing such graphs. Since this completion process can be reduced to an algorithm, as one solves $\sqrt{2}$ on a pocket calculator without thinking about it, one could, with the automation of the proposed framework, solve a complex CSE problem by drawing a diagram. Compared to traditional kriging, the proposed framework can be used to recover unknown functions with much scarcer data by exploiting interdependencies between multiple functions and variables. The underlying problem could therefore also be interpreted as a generalization of that of solving linear systems of equations to that of approximating unknown variables and functions with noisy, incomplete, and nonlinear dependencies. Numerous examples illustrate the flexibility, scope, efficacy, and robustness of the proposed framework and show how it can be used as a pathway to identifying simple solutions to classical CSE problems (digital twin modeling, dimension reduction, mode decomposition, etc.).

stat.ML↗