Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge independent kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent with geometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners.
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Processing 是一門開(kai)源編程語言和與之(zhi)配套的(de)集成開(kai)發環境（IDE）的(de)名稱(cheng)。Processing 在電子藝術(shu)和視覺設計(ji)社區(qu)被用來教授編程基礎(chu)，并運用于大量的(de)新媒體和互(hu)動藝術(shu)作品中。
Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given connectivity. We show that the collection of admissible configurations representable by such meshes form a smooth manifold. For this manifold of planar triangular meshes we propose a geodesically complete Riemannian metric. It is a distinguishing feature of this metric that it preserves the mesh connectivity and prevents the mesh from degrading along geodesic curves. We detail a symplectic numerical integrator for the geodesic equation in its Hamiltonian formulation. Numerical experiments show that the proposed metric keeps the cell aspect ratios bounded away from zero and thus avoids mesh degradation along arbitrarily long geodesic curves.
Reward design is a fundamental problem in reinforcement learning (RL). A misspecified or poorly designed reward can result in low sample efficiency and undesired behaviors. In this paper, we propose the idea of programmatic reward design, i.e. using programs to specify the reward functions in RL environments. Programs allow human engineers to express sub-goals and complex task scenarios in a structured and interpretable way. The challenge of programmatic reward design, however, is that while humans can provide the high-level structures, properly setting the low-level details, such as the right amount of reward for a specific sub-task, remains difficult. A major contribution of this paper is a probabilistic framework that can infer the best candidate programmatic reward function from expert demonstrations. Inspired by recent generative-adversarial approaches, our framework searches for the most likely programmatic reward function under which the optimally generated trajectories cannot be differentiated from the demonstrated trajectories. Experimental results show that programmatic reward functionslearned using this framework can significantly outperform those learned using existing reward learning algo-rithms, and enable RL agents to achieve state-of-the-artperformance on highly complex tasks.
Many fundamental problems in data mining can be reduced to one or more NP-hard combinatorial optimization problems. Recent advances in novel technologies such as quantum and quantum-inspired hardware promise a substantial speedup for solving these problems compared to when using general purpose computers but often require the problem to be modeled in a special form, such as an Ising or quadratic unconstrained binary optimization (QUBO) model, in order to take advantage of these devices. In this work, we focus on the important binary matrix factorization (BMF) problem which has many applications in data mining. We propose two QUBO formulations for BMF. We show how clustering constraints can easily be incorporated into these formulations. The special purpose hardware we consider is limited in the number of variables it can handle which presents a challenge when factorizing large matrices. We propose a sampling based approach to overcome this challenge, allowing us to factorize large rectangular matrices. In addition to these methods, we also propose a simple baseline algorithm which outperforms our more sophisticated methods in a few situations. We run experiments on the Fujitsu Digital Annealer, a quantum-inspired complementary metal-oxide-semiconductor (CMOS) annealer, on both synthetic and real data, including gene expression data. These experiments show that our approach is able to produce more accurate BMFs than competing methods.
Tensor optimization is crucial to massive machine learning and signal processing tasks. In this paper, we consider tensor optimization with a convex and well-conditioned objective function and reformulate it into a nonconvex optimization using the Burer-Monteiro type parameterization. We analyze the local convergence of applying vanilla gradient descent to the factored formulation and establish a local regularity condition under mild assumptions. We also provide a linear convergence analysis of the gradient descent algorithm started in a neighborhood of the true tensor factors. Complementary to the local analysis, this work also characterizes the global geometry of the best rank-one tensor approximation problem and demonstrates that for orthogonally decomposable tensors the problem has no spurious local minima and all saddle points are strict except for the one at zero which is a third-order saddle point.
In this paper, we use support vector machines (SVM) to develop a machine learning framework to discover phase space structures that distinguish between distinct reaction pathways. The SVM model is trained using data from trajectories of Hamilton's equations and works well even with relatively few trajectories. Moreover, this framework is specifically designed to require minimal a priori knowledge of the dynamics in a system. This makes our approach computationally better suited than existing methods for high-dimensional systems and systems where integrating trajectories is expensive. We benchmark our approach on Chesnavich's CH$_4^+$ Hamiltonian.
This paper introduces a new approach to inferring the second order properties of a multivariate log Gaussian Cox process (LGCP) with a complex intensity function. We assume a semi-parametric model for the multivariate intensity function containing an unspecified complex factor common to all types of points. Given this model we exploit the availability of several types of points to construct a second-order conditional composite likelihood to infer the pair correlation and cross pair correlation functions of the LGCP. Crucially this likelihood does not depend on the unspecified part of the intensity function. We also introduce a cross validation method for model selection and an algorithm for regularized inference that can be used to obtain sparse models for cross pair correlation functions. The methodology is applied to simulated data as well as data examples from microscopy and criminology. This shows how the new approach outperforms existing alternatives where the intensity functions are estimated non-parametrically.
With the development of data acquisition technology, multi-channel data is collected and widely used in many fields. Most of them can be expressed as various types of vector functions. Feature extraction of vector functions for identifying certain patterns of interest is a critical but challenging task. In this paper, we focus on constructing moment invariants of general vector functions. Specifically, we define rotation-affine transform to describe real deformations of general vector functions, and then design a structural frame to systematically generate Gaussian-Hermite moment invariants to this transform model. This is the first time that a uniform frame has been proposed in the literature to construct orthogonal moment invariants of general vector functions. Given a certain type of multi-channel data, we demonstrate how to utilize the new method to derive all possible invariants and to eliminate various dependences among them. For RGB images, 2D and 3D flow fields, we obtain the complete and independent sets of the invariants with low orders and low degrees. Based on synthetic and popular datasets of vector-valued data, the experiments are carried out to evaluate the stability and discriminability of these invariants, and also their robustness to noise. The results clearly show that the moment invariants proposed in our paper have better performance than other previously used moment invariants of vector functions in RGB image classification, vortex detection in 2D vector fields and template matching for 3D flow fields.
Kendall transformation is a conversion of an ordered feature into a vector of pairwise order relations between individual values. This way, it preserves ranking of observations and represents it in a categorical form. Such transformation allows for generalisation of methods requiring strictly categorical input, especially in the limit of small number of observations, when discretisation becomes problematic. In particular, many approaches of information theory can be directly applied to Kendall-transformed continuous data without relying on differential entropy or any additional parameters. Moreover, by filtering information to this contained in ranking, Kendall transformation leads to a better robustness at a reasonable cost of dropping sophisticated interactions which are anyhow unlikely to be correctly estimated. In bivariate analysis, Kendall transformation can be related to popular non-parametric methods, showing the soundness of the approach. The paper also demonstrates its efficiency in multivariate problems, as well as provides an example analysis of a real-world data.
The inductive biases of graph representation learning algorithms are often encoded in the background geometry of their embedding space. In this paper, we show that general directed graphs can be effectively represented by an embedding model that combines three components: a pseudo-Riemannian metric structure, a non-trivial global topology, and a unique likelihood function that explicitly incorporates a preferred direction in embedding space. We demonstrate the representational capabilities of this method by applying it to the task of link prediction on a series of synthetic and real directed graphs from natural language applications and biology. In particular, we show that low-dimensional cylindrical Minkowski and anti-de Sitter spacetimes can produce equal or better graph representations than curved Riemannian manifolds of higher dimensions.
We study object recognition under the constraint that each object class is only represented by very few observations. In such cases, naive supervised learning would lead to severe over-fitting in deep neural networks due to limited training data. We tackle this problem by creating much more training data through label propagation from the few labeled examples to a vast collection of unannotated images. Our main insight is that such a label propagation scheme can be highly effective when the similarity metric used for propagation is learned and transferred from other related domains with lots of data. We test our approach on semi-supervised learning, transfer learning and few-shot recognition, where we learn our similarity metric using various supervised/unsupervised pretraining methods, and transfer it to unlabeled data across different data distributions. By taking advantage of unlabeled data in this way, we achieve significant improvements on all three tasks. Notably, our approach outperforms current state-of-the-art techniques by an absolute $20\%$ for semi-supervised learning on CIFAR10, $10\%$ for transfer learning from ImageNet to CIFAR10, and $6\%$ for few-shot recognition on mini-ImageNet, when labeled examples are limited.
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