2,651 research outputs found
Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph
Data-sensitive metrics adapt distances locally based the density of data
points with the goal of aligning distances and some notion of similarity. In
this paper, we give the first exact algorithm for computing a data-sensitive
metric called the nearest neighbor metric. In fact, we prove the surprising
result that a previously published -approximation is an exact algorithm.
The nearest neighbor metric can be viewed as a special case of a
density-based distance used in machine learning, or it can be seen as an
example of a manifold metric. Previous computational research on such metrics
despaired of computing exact distances on account of the apparent difficulty of
minimizing over all continuous paths between a pair of points. We leverage the
exact computation of the nearest neighbor metric to compute sparse spanners and
persistent homology. We also explore the behavior of the metric built from
point sets drawn from an underlying distribution and consider the more general
case of inputs that are finite collections of path-connected compact sets.
The main results connect several classical theories such as the conformal
change of Riemannian metrics, the theory of positive definite functions of
Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop
novel proof techniques based on the combination of screw functions and
Lipschitz extensions that may be of independent interest.Comment: 15 page
A Novel Approach to Face Recognition using Image Segmentation based on SPCA-KNN Method
In this paper we propose a novel method for face recognition using hybrid SPCA-KNN (SIFT-PCA-KNN) approach. The proposed method consists of three parts. The first part is based on preprocessing face images using Graph Based algorithm and SIFT (Scale Invariant Feature Transform) descriptor. Graph Based topology is used for matching two face images. In the second part eigen values and eigen vectors are extracted from each input face images. The goal is to extract the important information from the face data, to represent it as a set of new orthogonal variables called principal components. In the final part a nearest neighbor classifier is designed for classifying the face images based on the SPCA-KNN algorithm. The algorithm has been tested on 100 different subjects (15 images for each class). The experimental result shows that the proposed method has a positive effect on overall face recognition performance and outperforms other examined methods
Linear-Size Approximations to the Vietoris-Rips Filtration
The Vietoris-Rips filtration is a versatile tool in topological data
analysis. It is a sequence of simplicial complexes built on a metric space to
add topological structure to an otherwise disconnected set of points. It is
widely used because it encodes useful information about the topology of the
underlying metric space. This information is often extracted from its so-called
persistence diagram. Unfortunately, this filtration is often too large to
construct in full. We show how to construct an O(n)-size filtered simplicial
complex on an -point metric space such that its persistence diagram is a
good approximation to that of the Vietoris-Rips filtration. This new filtration
can be constructed in time. The constant factors in both the size
and the running time depend only on the doubling dimension of the metric space
and the desired tightness of the approximation. For the first time, this makes
it computationally tractable to approximate the persistence diagram of the
Vietoris-Rips filtration across all scales for large data sets.
We describe two different sparse filtrations. The first is a zigzag
filtration that removes points as the scale increases. The second is a
(non-zigzag) filtration that yields the same persistence diagram. Both methods
are based on a hierarchical net-tree and yield the same guarantees
A random-permutations-based approach to fast read alignment
BACKGROUND: Read alignment is a computational bottleneck in some sequencing projects. Most of the existing software packages for read alignment are based on two algorithmic approaches: prefix-trees and hash-tables. We propose a new approach to read alignment using random permutations of strings. RESULTS: We present a prototype implementation and experiments performed with simulated and real reads of human DNA. Our experiments indicate that this permutations-based prototype is several times faster than comparable programs for fast read alignment and that it aligns more reads correctly. CONCLUSIONS: This approach may lead to improved speed, sensitivity, and accuracy in read alignment. The algorithm can also be used for specialized alignment applications and it can be extended to other related problems, such as assembly. More information: http://alignment.commons.yale.ed
- …