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Register a new userA Multiscale Environment for Learning by Diffusion
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Sam Polk
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Clustering algorithms partition a dataset into groups of similar points. The clustering problem is very general, and different partitions of the same dataset could be considered correct and useful. To fully understand such data, it must be considered at a variety of scales, ranging from coarse to fine. We introduce the Multiscale Environment for Learning by Diffusion (MELD) data model, which is a family of clusterings parameterized by nonlinear diffusion on the dataset. We show that the MELD data model precisely captures latent multiscale structure in data and facilitates its analysis. To efficiently learn the multiscale structure observed in many real datasets, we introduce the Multiscale Learning by Unsupervised Nonlinear Diffusion (M-LUND) clustering algorithm, which is derived from a diffusion process at a range of temporal scales. We provide theoretical guarantees for the algorithms performance and establish its computational efficiency. Finally, we show that the M-LUND clustering algorithm detects the latent structure in a range of synthetic and real datasets.
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Clustering algorithms partition a dataset into groups of similar points. The primary contribution of this article is the Multiscale Spatially-Regularized Diffusion Learning (M-SRDL) clustering algorithm, which uses spatially-regularized diffusion distances to efficiently and accurately learn multiple scales of latent structure in hyperspectral images (HSI). The M-SRDL clustering algorithm extracts clusterings at many scales from an HSI and outputs these clusterings variation of information-barycenter as an exemplar for all underlying cluster structure. We show that incorporating spatial regularization into a multiscale clustering framework corresponds to smoother and more coherent clusters when applied to HSI data and leads to more accurate clustering labels.
Machine learning methods have greatly changed science, engineering, finance, business, and other fields. Despite the tremendous accomplishments of machine learning and deep learning methods, many challenges still remain. In particular, the performance of machine learning methods is often severely affected in case of diverse data, usually associated with smaller data sets or data related to areas of study where the size of the data sets is constrained by the complexity and/or high cost of experiments. Moreover, data with limited labeled samples is a challenge to most learning approaches. In this paper, the aforementioned challenges are addressed by integrating graph-based frameworks, multiscale structure, modified and adapted optimization procedures and semi-supervised techniques. This results in two innovative multiscale Laplacian learning (MLL) approaches for machine learning tasks, such as data classification, and for tackling diverse data, data with limited samples and smaller data sets. The first approach, called multikernel manifold learning (MML), integrates manifold learning with multikernel information and solves a regularization problem consisting of a loss function and a warped kernel regularizer using multiscale graph Laplacians. The second approach, called the multiscale MBO (MMBO) method, introduces multiscale Laplacians to a modification of the famous classical Merriman-Bence-Osher (MBO) scheme, and makes use of fast solvers for finding the approximations to the extremal eigenvectors of the graph Laplacian. We demonstrate the performance of our methods experimentally on a variety of data sets, such as biological, text and image data, and compare them favorably to existing approaches.
In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant. Interactive theorem provers such as Coq enable users to construct machine-checkable proofs in a step-by-step manner. Hence, they provide an opportunity to explore theorem proving with human supervision. We use GamePad to synthesize proofs for a simple algebraic rewrite problem and train baseline models for a formalization of the Feit-Thompson theorem. We address position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict the next proof step) tasks, which arise naturally in tactic-based theorem proving.
We propose a new class of implicit networks, the multiscale deep equilibrium model (MDEQ), suited to large-scale and highly hierarchical pattern recognition domains. An MDEQ directly solves for and backpropagates through the equilibrium points of multiple feature resolutions simultaneously, using implicit differentiation to avoid storing intermediate states (and thus requiring only $O(1)$ memory consumption). These simultaneously-learned multi-resolution features allow us to train a single model on a diverse set of tasks and loss functions, such as using a single MDEQ to perform both image classification and semantic segmentation. We illustrate the effectiveness of this approach on two large-scale vision tasks: ImageNet classification and semantic segmentation on high-resolution images from the Cityscapes dataset. In both settings, MDEQs are able to match or exceed the performance of recent competitive computer vision models: the first time such performance and scale have been achieved by an implicit deep learning approach. The code and pre-trained models are at https://github.com/locuslab/mdeq .
Physical processes influencing the properties of galaxies can be traced by the dependence and evolution of galaxy properties on their environment. A detailed understanding of this dependence can only be gained through comparison of observations with models, with an appropriate quantification of the rich parameter space describing the environment of the galaxy. We present a new, multiscale parameterization of galaxy environment which retains an observationally motivated simplicity whilst utilizing the information present on different scales. We examine how the distribution of galaxy (u-r) colours in the Sloan Digital Sky Survey (SDSS), parameterized using a double gaussian (red plus blue peak) fit, depends upon multiscale density. This allows us to probe the detailed dependence of galaxy properties on environment in a way which is independent of the halo model. Nonetheless, cross-correlation with the group catalogue constructed by Yang et al, 2007 shows that galaxy properties trace environment on different scales in a way which mimics that expected within the halo model. This provides independent support for the existence of virialized haloes, and important additional clues to the role played by environment in the evolution of the galaxy population. This work is described in full by Wilman et al., 2010, MNRAS, accepted
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