89,861 research outputs found
Positive semi-definite embedding for dimensionality reduction and out-of-sample extensions
In machine learning or statistics, it is often desirable to reduce the
dimensionality of a sample of data points in a high dimensional space
. This paper introduces a dimensionality reduction method where
the embedding coordinates are the eigenvectors of a positive semi-definite
kernel obtained as the solution of an infinite dimensional analogue of a
semi-definite program. This embedding is adaptive and non-linear. A main
feature of our approach is the existence of a non-linear out-of-sample
extension formula of the embedding coordinates, called a projected Nystr\"om
approximation. This extrapolation formula yields an extension of the kernel
matrix to a data-dependent Mercer kernel function. Our empirical results
indicate that this embedding method is more robust with respect to the
influence of outliers, compared with a spectral embedding method.Comment: 16 pages, 5 figures. Improved presentatio
Fast Robust PCA on Graphs
Mining useful clusters from high dimensional data has received significant
attention of the computer vision and pattern recognition community in the
recent years. Linear and non-linear dimensionality reduction has played an
important role to overcome the curse of dimensionality. However, often such
methods are accompanied with three different problems: high computational
complexity (usually associated with the nuclear norm minimization),
non-convexity (for matrix factorization methods) and susceptibility to gross
corruptions in the data. In this paper we propose a principal component
analysis (PCA) based solution that overcomes these three issues and
approximates a low-rank recovery method for high dimensional datasets. We
target the low-rank recovery by enforcing two types of graph smoothness
assumptions, one on the data samples and the other on the features by designing
a convex optimization problem. The resulting algorithm is fast, efficient and
scalable for huge datasets with O(nlog(n)) computational complexity in the
number of data samples. It is also robust to gross corruptions in the dataset
as well as to the model parameters. Clustering experiments on 7 benchmark
datasets with different types of corruptions and background separation
experiments on 3 video datasets show that our proposed model outperforms 10
state-of-the-art dimensionality reduction models. Our theoretical analysis
proves that the proposed model is able to recover approximate low-rank
representations with a bounded error for clusterable data
Semi-Supervised Discriminant Analysis Using Robust Path-Based Similarity
Linear Discriminant Analysis (LDA), which works by maximizing the within-class similarity and minimizing the between-class similarity simultaneously, is a popular dimensionality reduction technique in pattern recognition and machine learning. In real-world applications when labeled data are limited, LDA does not work well. Under many situations, however, it is easy to obtain unlabeled data in large quantities. In this paper, we propose a novel dimensionality reduction method, called Semi-Supervised Discriminant Analysis (SSDA), which can utilize both labeled and unlabeled data to perform dimensionality reduction in the semisupervised setting. Our method uses a robust path-based similarity measure to capture the manifold structure of the data and then uses the obtained similarity to maximize the separability between different classes. A kernel extension of the proposed method for nonlinear dimensionality reduction in the semi-supervised setting is also presented. Experiments on face recognition demonstrate the effectiveness of the proposed method. 1
Compressed matched filter for non-Gaussian noise
We consider estimation of a deterministic unknown parameter vector in a
linear model with non-Gaussian noise. In the Gaussian case, dimensionality
reduction via a linear matched filter provides a simple low dimensional
sufficient statistic which can be easily communicated and/or stored for future
inference. Such a statistic is usually unknown in the general non-Gaussian
case. Instead, we propose a hybrid matched filter coupled with a randomized
compressed sensing procedure, which together create a low dimensional
statistic. We also derive a complementary algorithm for robust reconstruction
given this statistic. Our recovery method is based on the fast iterative
shrinkage and thresholding algorithm which is used for outlier rejection given
the compressed data. We demonstrate the advantages of the proposed framework
using synthetic simulations
Robust Principal Component Analysis on Graphs
Principal Component Analysis (PCA) is the most widely used tool for linear
dimensionality reduction and clustering. Still it is highly sensitive to
outliers and does not scale well with respect to the number of data samples.
Robust PCA solves the first issue with a sparse penalty term. The second issue
can be handled with the matrix factorization model, which is however
non-convex. Besides, PCA based clustering can also be enhanced by using a graph
of data similarity. In this article, we introduce a new model called "Robust
PCA on Graphs" which incorporates spectral graph regularization into the Robust
PCA framework. Our proposed model benefits from 1) the robustness of principal
components to occlusions and missing values, 2) enhanced low-rank recovery, 3)
improved clustering property due to the graph smoothness assumption on the
low-rank matrix, and 4) convexity of the resulting optimization problem.
Extensive experiments on 8 benchmark, 3 video and 2 artificial datasets with
corruptions clearly reveal that our model outperforms 10 other state-of-the-art
models in its clustering and low-rank recovery tasks
Maximum Entropy Linear Manifold for Learning Discriminative Low-dimensional Representation
Representation learning is currently a very hot topic in modern machine
learning, mostly due to the great success of the deep learning methods. In
particular low-dimensional representation which discriminates classes can not
only enhance the classification procedure, but also make it faster, while
contrary to the high-dimensional embeddings can be efficiently used for visual
based exploratory data analysis.
In this paper we propose Maximum Entropy Linear Manifold (MELM), a
multidimensional generalization of Multithreshold Entropy Linear Classifier
model which is able to find a low-dimensional linear data projection maximizing
discriminativeness of projected classes. As a result we obtain a linear
embedding which can be used for classification, class aware dimensionality
reduction and data visualization. MELM provides highly discriminative 2D
projections of the data which can be used as a method for constructing robust
classifiers.
We provide both empirical evaluation as well as some interesting theoretical
properties of our objective function such us scale and affine transformation
invariance, connections with PCA and bounding of the expected balanced accuracy
error.Comment: submitted to ECMLPKDD 201
Robust 2D Joint Sparse Principal Component Analysis with F-Norm Minimization for Sparse Modelling: 2D-RJSPCA
© 2018 IEEE. Principal component analysis (PCA) is widely used methods for dimensionality reduction and Lots of variants have been proposed to improve the robustness of algorithm, however, these methods suffer from the fact that PCA is linear combination which makes it difficult to interpret complex nonlinear data, and sensitive to outliers or cannot extract features consistently, i.e., collectively; PCA may still require measuring all input features. 2DPCA based on 1-norm has been recently used for robust dimensionality reduction in the image domain but still sensitive to noise. In this paper, we introduce robust formation of 2DPCA by centering the data using the optimized mean for two-dimensional joint sparse as well as effectively combining the robustness of 2DPCA and the sparsity-inducing lasso regularization. Optimal mean helps to improve the robustness of joint sparse PCA further. The distance in spatial dimension is measure in F-norm and sum of different datapoint uses 1-norm. 2DR-JSPCA imposes joint sparse constraints on its objective function whereas additional plenty term help to deal with outliers efficiently. Both theoretical and empirical results on six publicly available benchmark datasets shows that Optimal mean 2DR-JSPCA provides better performance for dimensionality reduction as compare to non-sparse (2DPCA and 2DPCA-L1) and sparse (SPCA, JSPCA)
Manifold Based Deep Learning: Advances and Machine Learning Applications
Manifolds are topological spaces that are locally Euclidean and find applications in dimensionality reduction, subspace learning, visual domain adaptation, clustering, and more. In this dissertation, we propose a framework for linear dimensionality reduction called the proxy matrix optimization (PMO) that uses the Grassmann manifold for optimizing over orthogonal matrix manifolds. PMO is an iterative and flexible method that finds the lower-dimensional projections for various linear dimensionality reduction methods by changing the objective function. PMO is suitable for Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), Canonical Correlation Analysis (CCA), Maximum Autocorrelation Factors (MAF), and Locality Preserving Projections (LPP). We extend PMO to incorporate robust Lp-norm versions of PCA and LDA, which uses fractional p-norms making them more robust to noisy data and outliers. The PMO method is designed to be realized as a layer in a neural network for maximum benefit. In order to do so, the incremental versions of PCA, LDA, and LPP are included in the PMO framework for problems where the data is not all available at once. Next, we explore the topic of domain shift in visual domain adaptation by combining concepts from spherical manifolds and deep learning. We investigate domain shift, which quantifies how well a model trained on a source domain adapts to a similar target domain with a metric called Spherical Optimal Transport (SpOT). We adopt the spherical manifold along with an orthogonal projection loss to obtain the features from the source and target domains. We then use the optimal transport with the cosine distance between the features as a way to measure the gap between the domains. We show, in our experiments with domain adaptation datasets, that SpOT does better than existing measures for quantifying domain shift and demonstrates a better correlation with the gain of transfer across domains
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