2,797 research outputs found
DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling
This paper explores a fully unsupervised deep learning approach for computing
distance-preserving maps that generate low-dimensional embeddings for a certain
class of manifolds. We use the Siamese configuration to train a neural network
to solve the problem of least squares multidimensional scaling for generating
maps that approximately preserve geodesic distances. By training with only a
few landmarks, we show a significantly improved local and nonlocal
generalization of the isometric mapping as compared to analogous non-parametric
counterparts. Importantly, the combination of a deep-learning framework with a
multidimensional scaling objective enables a numerical analysis of network
architectures to aid in understanding their representation power. This provides
a geometric perspective to the generalizability of deep learning.Comment: 10 pages, 11 Figure
Construction of embedded fMRI resting state functional connectivity networks using manifold learning
We construct embedded functional connectivity networks (FCN) from benchmark
resting-state functional magnetic resonance imaging (rsfMRI) data acquired from
patients with schizophrenia and healthy controls based on linear and nonlinear
manifold learning algorithms, namely, Multidimensional Scaling (MDS), Isometric
Feature Mapping (ISOMAP) and Diffusion Maps. Furthermore, based on key global
graph-theoretical properties of the embedded FCN, we compare their
classification potential using machine learning techniques. We also assess the
performance of two metrics that are widely used for the construction of FCN
from fMRI, namely the Euclidean distance and the lagged cross-correlation
metric. We show that the FCN constructed with Diffusion Maps and the lagged
cross-correlation metric outperform the other combinations
Outlier Mining Methods Based on Graph Structure Analysis
Outlier detection in high-dimensional datasets is a fundamental and challenging problem across disciplines that has also practical implications, as removing outliers from the training set improves the performance of machine learning algorithms. While many outlier mining algorithms have been proposed in the literature, they tend to be valid or efficient for specific types of datasets (time series, images, videos, etc.). Here we propose two methods that can be applied to generic datasets, as long as there is a meaningful measure of distance between pairs of elements of the dataset. Both methods start by defining a graph, where the nodes are the elements of the dataset, and the links have associated weights that are the distances between the nodes. Then, the first method assigns an outlier score based on the percolation (i.e., the fragmentation) of the graph. The second method uses the popular IsoMap non-linear dimensionality reduction algorithm, and assigns an outlier score by comparing the geodesic distances with the distances in the reduced space. We test these algorithms on real and synthetic datasets and show that they either outperform, or perform on par with other popular outlier detection methods. A main advantage of the percolation method is that is parameter free and therefore, it does not require any training; on the other hand, the IsoMap method has two integer number parameters, and when they are appropriately selected, the method performs similar to or better than all the other methods tested.Peer ReviewedPostprint (published version
Masking Strategies for Image Manifolds
We consider the problem of selecting an optimal mask for an image manifold,
i.e., choosing a subset of the pixels of the image that preserves the
manifold's geometric structure present in the original data. Such masking
implements a form of compressive sensing through emerging imaging sensor
platforms for which the power expense grows with the number of pixels acquired.
Our goal is for the manifold learned from masked images to resemble its full
image counterpart as closely as possible. More precisely, we show that one can
indeed accurately learn an image manifold without having to consider a large
majority of the image pixels. In doing so, we consider two masking methods that
preserve the local and global geometric structure of the manifold,
respectively. In each case, the process of finding the optimal masking pattern
can be cast as a binary integer program, which is computationally expensive but
can be approximated by a fast greedy algorithm. Numerical experiments show that
the relevant manifold structure is preserved through the data-dependent masking
process, even for modest mask sizes
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