140 research outputs found
Bayesian Inference on Matrix Manifolds for Linear Dimensionality Reduction
We reframe linear dimensionality reduction as a problem of Bayesian inference
on matrix manifolds. This natural paradigm extends the Bayesian framework to
dimensionality reduction tasks in higher dimensions with simpler models at
greater speeds. Here an orthogonal basis is treated as a single point on a
manifold and is associated with a linear subspace on which observations vary
maximally. Throughout this paper, we employ the Grassmann and Stiefel manifolds
for various dimensionality reduction problems, explore the connection between
the two manifolds, and use Hybrid Monte Carlo for posterior sampling on the
Grassmannian for the first time. We delineate in which situations either
manifold should be considered. Further, matrix manifold models are used to
yield scientific insight in the context of cognitive neuroscience, and we
conclude that our methods are suitable for basic inference as well as accurate
prediction.Comment: All datasets and computer programs are publicly available at
http://www.ics.uci.edu/~babaks/Site/Codes.htm
Linear dimensionality reduction: Survey, insights, and generalizations
Linear dimensionality reduction methods are a cornerstone of analyzing high
dimensional data, due to their simple geometric interpretations and typically
attractive computational properties. These methods capture many data features
of interest, such as covariance, dynamical structure, correlation between data
sets, input-output relationships, and margin between data classes. Methods have
been developed with a variety of names and motivations in many fields, and
perhaps as a result the connections between all these methods have not been
highlighted. Here we survey methods from this disparate literature as
optimization programs over matrix manifolds. We discuss principal component
analysis, factor analysis, linear multidimensional scaling, Fisher's linear
discriminant analysis, canonical correlations analysis, maximum autocorrelation
factors, slow feature analysis, sufficient dimensionality reduction,
undercomplete independent component analysis, linear regression, distance
metric learning, and more. This optimization framework gives insight to some
rarely discussed shortcomings of well-known methods, such as the suboptimality
of certain eigenvector solutions. Modern techniques for optimization over
matrix manifolds enable a generic linear dimensionality reduction solver, which
accepts as input data and an objective to be optimized, and returns, as output,
an optimal low-dimensional projection of the data. This simple optimization
framework further allows straightforward generalizations and novel variants of
classical methods, which we demonstrate here by creating an
orthogonal-projection canonical correlations analysis. More broadly, this
survey and generic solver suggest that linear dimensionality reduction can move
toward becoming a blackbox, objective-agnostic numerical technology.JPC and ZG received funding from the UK Engineering and Physical Sciences Research Council (EPSRC EP/H019472/1). JPC received funding from a Sloan Research Fellowship, the Simons Foundation (SCGB#325171 and SCGB#325233), the Grossman Center at Columbia University, and the Gatsby Charitable Trust.This is the author accepted manuscript. The final version is available from MIT Press via http://jmlr.org/papers/v16/cunningham15a.htm
階層型神経回路モデルにおける学習力学の幾何学的理論
学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 岡田 真人, 東京大学教授 津田 宏治, 東京大学教授 能瀬 聡直, 東京大学准教授 國廣 昇, 東京大学講師 佐藤 一誠University of Tokyo(東京大学
Discriminative Video Representation Learning
Representation learning is a fundamental research problem in the area of machine learning, refining the raw data to discover representations needed for various applications. However, real-world data, particularly video data, is neither mathematically nor computationally convenient to process due to its semantic redundancy and complexity. Video data, as opposed to images, includes temporal correlation and motion dynamics, but the ground truth label is normally limited to category labels, which makes the video representation learning a challenging problem. To this end, this thesis addresses the problem of video representation learning, specifically discriminative video representation learning, which focuses on capturing useful data distributions and reliable feature representations improving the performance of varied downstream tasks. We argue that neither all frames in one video nor all dimensions in one feature vector are useful and should be equally treated for video representation learning. Based on this argument, several novel algorithms are investigated in this thesis under multiple application scenarios, such as action recognition, action detection and one-class video anomaly detection. These proposed video representation learning methods produce discriminative video features in both deep and non-deep learning setups. Specifically, they are presented in the form of: 1) an early fusion layer that adopts a temporal ranking SVM formulation, agglomerating several optical flow images from consecutive frames into a novel compact representation, named as dynamic optical flow images; 2) an intermediate feature aggregation layer that applies weakly-supervised contrastive learning techniques, learning discriminative video representations via contrasting positive and negative samples from a sequence; 3) a new formulation for one-class feature learning that learns a set of discriminative subspaces with orthonormal hyperplanes to flexibly bound the one-class data distribution using Riemannian optimisation methods. We provide extensive experiments to gain intuitions into why the learned representations are discriminative and useful. All the proposed methods in this thesis are evaluated on standard publicly available benchmarks, demonstrating state-of-the-art performance
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Identification and Validation of Structures in Neural Population Responses
A fundamental challenge of neuroscience is to understand how interconnected populations of neurons give rise to the remarkable computational abilities of our brains. Large neural datasets offer promise, but they are perilous: they are too complex to be studied with traditional single-neuron analyses, and thus require new analyses that can uncover structure at the level of the population. However, since these analyses operate on large datasets, our intuition whether structure is significant breaks down. Hence, we run the risk of over-interpreting structure from the population data that may have a simple explanation. Thus, with population analysis methods, there is also a need for methods that can validate the significance of structure identified. In this dissertation, I discuss topics covering both the identification and the validation of structure in population data. In the first part, I discuss novel methods for uncovering the computational strategy employed by the motor cortex to flexibly switch between different neural computations. I demonstrate that collective activity patterns of motor cortex neurons related to different computations are orthogonal yet can still be linked, indicating a degree of flexibility that was not displayed or predicted by existing cortical models. In the second part, I discuss a novel analytical framework to rigorously test the novelty of population-level findings, given a specified set of primary features such as correlations across time, neurons and experimental conditions. This framework provides a general tool for validating population findings across the brain and across population-level hypotheses
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