Kernel matrix-vector multiplication (KMVM) is a foundational operation in machine learning and scientific computing. However, as KMVM tends to scale quadratically in both memory and time, applications are often limited by these computational constraints. In this paper, we propose a novel approximation procedure coined \textit{Faster-Fast and Free Memory Method} ($\text{F}^3$M) to address these scaling issues of KMVM for tall~($10^8\sim 10^9$) and skinny~($D\leq7$) data. Extensive experiments demonstrate that $\text{F}^3$M has empirical \emph{linear time and memory} complexity with a relative error of order $10^{-3}$ and can compute a full KMVM for a billion points \emph{in under a minute} on a high-end GPU, leading to a significant speed-up in comparison to existing CPU methods. We demonstrate the utility of our procedure by applying it as a drop-in for the state-of-the-art GPU-based linear solver FALKON, \emph{improving speed 1.5-5.5 times} at the cost of $<1\%$ drop in accuracy. We further demonstrate competitive results on \emph{Gaussian Process regression} coupled with significant speedups on a variety of real-world datasets.

The KeOps library provides a fast and memory-efficient GPU support for tensors whose entries are given by a mathematical formula, such as kernel and distance matrices. KeOps alleviates the main bottleneck of tensor-centric libraries for kernel and geometric applications: memory consumption. It also supports automatic differentiation and outperforms standard GPU baselines, including PyTorch CUDA tensors or the Halide and TVM libraries. KeOps combines optimized C++/CUDA schemes with binders for high-level languages: Python (Numpy and PyTorch), Matlab and GNU R. As a result, high-level “quadratic” codes can now scale up to large data sets with millions of samples processed in seconds. KeOps brings graphics-like performances for kernel methods and is freely available on standard repositories (PyPi, CRAN). To showcase its versatility, we provide tutorials in a wide range of settings online at www.kernel-operations.io.

Geometric methods rely on tensors that can be encoded using a symbolic formula and data arrays, such as kernel and distance matrices. We present an extension for standard machine learning frameworks that provides comprehensive support for this abstraction on CPUs and GPUs: our toolbox combines a versatile, transparent user interface with fast runtimes and low memory usage. Unlike general purpose acceleration frameworks such as XLA, our library turns generic Python code into binaries whose performances are competitive with state-of-the-art geometric libraries - such as FAISS for nearest neighbor search - with the added benefit of flexibility. We perform an extensive evaluation on a broad class of problems: Gaussian modelling, K-nearest neighbors search, geometric deep learning, non-Euclidean embeddings and optimal transport theory. In practice, for geometric problems that involve 1k to 1M samples in dimension 1 to 100, our library speeds up baseline GPU implementations by up to two orders of magnitude.

This paper provides some considerations regarding using individualized head-related transfer functions for rendering binaural spatial audio over headphones. It briefly considers the degree of benefit that individualization may provide. It then examines the degree of variation existing within the ear morphology across listeners within the Sydney-York Morphological and Recording of Ears (SYMARE) database using kernel principal component analysis and the large deformation diffeomorphic metric mapping framework. The degree of variation across listeners in the directivity patterns associated with head-related transfer functions is also analyzed as a function of frequency. The variation in ear morphology is related to the variation in the directivity patterns using simple linear regression.

This paper describes features in the ear shape that change across a population of ears and explores the corresponding changes in ear acoustics. The statistical analysis conducted over the space of ear shapes uses a kernel principal component analysis (KPCA). Further, it utilizes the framework of large deformation diffeomorphic metric mapping and the vector space that is constructed over the space of initial momentums, which describes the diffeomorphic transformations from the reference template ear shape. The population of ear shapes examined by the KPCA are 124 left and right ear shapes from the SYMARE database that were rigidly aligned to the template (population average) ear. In the work presented here we show the morphological variations captured by the first two kernel principal components, and also show the acoustic transfer functions of the ears which are computed using fast multipole boundary element method simulations.

This paper describes the generation of a morphable model for external ear shapes. The aim for the morphable model is to characterize an ear shape using only a few parameters in order to assist the study of morphoacoustics. The model is derived from a statistical analysis of a population of 58 ears from the SYMARE database. It is based upon the framework of large deformation diffeomorphic metric mapping (LDDMM) and the vector space that is constructed over the space of initial momentums describing the diffeomorphic transformations. To develop a morphable model using the LDDMM framework, the initial momentums are analyzed using a kernel based principal component analysis. In this paper, we examine the ability of our morphable model to construct test ear shapes not included in the principal component analysis.

In this work we introduce a new dissimilarity measure for shape registration using the notion of normal cycles, a concept from geometric measure theory which allows us to generalize curvature for nonsmooth subsets of the Euclidean space. Our construction is based on the definition of kernel metrics on the space of normal cycles which take explicit expressions in a discrete setting. This approach is closely similar to previous works based on currents and varifolds [M. Vaillant and J. Glaunès, Surface matching via currents, in Information Processing in Medical Imaging, G. E. Christensen and M. Sonka, eds., Lecture Notes in Comput. Sci. 3565, Springer, Berlin, 2005, pp. 381--392; N. Charon and A. Trouvé, SIAM J. Imaging Sci., 6 (2013), pp. 2547--2580]. We derive the computational setting for discrete curves in $\mathbb{R}^3$, using the large deformation diffeomorphic metric mapping framework as the model for deformations. We present synthetic and real data experiments and compare them with the currents and varifolds approaches.

This paper describes how Large Deformation Diffeomorphic Metric Mapping (LDDMM) can be coupled with a Fast Multipole (FM) Boundary Element Method (BEM) to investigate the relationship between morphological changes in the head, torso, and outer ears and their acoustic filtering (described by Head Related Transfer Functions, HRTFs). The LDDMM technique provides the ability to study and implement morphological changes in ear, head and torso shapes. The FM-BEM technique provides numerical simulations of the acoustic properties of an individual's head, torso, and outer ears. This paper describes the first application of LDDMM to the study of the relationship between a listener's morphology and a listener's HRTFs. To demonstrate some of the new capabilities provided by the coupling of these powerful tools, we morph the shape of a listener's ear, while keeping the torso and head shape essentially constant, and show changes in the acoustics. We validate the methodological framework by mapping the complete morphology of one listener to a target listener and obtaining the target listener's HRTFs. This work utilizes the data provided by the Sydney York Morphological and Acoustic Recordings of Ears (SYMARE) database.

This paper presents a framework to introduce spatial and anatomical priors in SVM for brain image analysis based on regularization operators. A notion of proximity based on prior anatomical knowledge between the image points is defined by a graph (e.g., brain connectivity graph) or a metric (e.g., Fisher metric on statistical manifolds). A regularization operator is then defined from the graph Laplacian, in the discrete case, or from the Laplace-Beltrami operator, in the continuous case. The regularization operator is then introduced into the SVM, which exponentially penalizes high-frequency components with respect to the graph or to the metric and thus constrains the classification function to be smooth with respect to the prior. It yields a new SVM optimization problem whose kernel is a heat kernel on graphs or on manifolds. We then present different types of priors and provide efficient computations of the Gram matrix. The proposed framework is finally applied to the classification of brain Magnetic Resonance (MR) images (based on Gray Matter (GM) concentration maps and cortical thickness measures) from 137 patients with Alzheimer's Disease (AD) and 162 elderly controls. The results demonstrate that the proposed classifier generates less-noisy and consequently more interpretable feature maps with high classification performances.

The alignment and normalization of individual brain structures is a prerequisite for group-level analyses of structural and functional neuroimaging data. The techniques currently available are either based on volume and/or surface attributes, with limited insight regarding the consistent alignment of anatomical landmarks across individuals. This article details a global, geometric approach that performs the alignment of the exhaustive sulcal imprints (cortical folding patterns) across individuals. This DIffeomorphic Sulcal-based COrtical (DISCO) technique proceeds to the automatic extraction, identification and simplification of sulcal features from T1-weighted Magnetic Resonance Image (MRI) series. These features are then used as control measures for fully-3-D diffeomorphic deformations. Quantitative and qualitative evaluations show that DISCO correctly aligns the sulcal folds and gray and white matter volumes across individuals. The comparison with a recent, iconic diffeomorphic approach (DARTEL) highlights how the absence of explicit cortical landmarks may lead to the misalignment of cortical sulci. We also feature DISCO in the automatic design of an empirical sulcal template from group data. We also demonstrate how DISCO can efficiently be combined with an image-based deformation (DARTEL) to further improve the consistency and accuracy of alignment performances. Finally, we illustrate how the optimized alignment of cortical folds across subjects improves sensitivity in the detection of functional activations in a group-level analysis of neuroimaging data.