### abstract ###
The emergence of low-cost sensor architectures for diverse modalities has made it possible to deploy sensor arrays that capture a single event from a large number of vantage points and using multiple modalities
In many scenarios, these sensors acquire very high-dimensional data such as audio signals, images, and video
To cope with such high-dimensional data, we typically rely on low-dimensional models
Manifold models provide a particularly powerful model that captures the structure of high-dimensional data when it is governed by a low-dimensional set of parameters
However, these models do not typically take into account dependencies among multiple sensors
We thus propose a new  joint manifold  framework for data ensembles that exploits such dependencies
We show that simple algorithms can exploit the joint manifold structure to improve their performance on standard signal processing applications
Additionally, recent results concerning dimensionality reduction for manifolds enable us to formulate a network-scalable data compression scheme that uses random projections of the sensed data
This scheme efficiently fuses the data from all sensors through the addition of such projections, regardless of the data modalities and dimensions
### introduction ###
The geometric notion of a low-dimensional manifold is a common, yet powerful, tool for modeling high-dimensional data
Manifold models arise in cases where ( i ) a  SYMBOL -dimensional parameter  SYMBOL  can be identified that carries the relevant information about a signal and ( ii ) the signal  SYMBOL  changes as a continuous (typically nonlinear) function of these parameters
Some typical examples include a one-dimensional (1-D) signal shifted by an unknown time delay (parameterized by the translation variable), a recording of a speech signal (parameterized by the underlying phonemes spoken by the speaker), and an image of a 3-D object at an unknown location captured from an unknown viewing angle (parameterized by the 3-D coordinates of the object and its roll, pitch, and yaw)
In these and many other cases, the geometry of the signal class forms a nonlinear  SYMBOL -dimensional manifold in  SYMBOL ,  SYMBOL } where  SYMBOL  is the  SYMBOL -dimensional parameter space~ CITATION
Low-dimensional manifolds have also been proposed as approximate models for nonparametric signal classes such as images of human faces or handwritten digits~ CITATION
In many scenarios, multiple observations of the same event may be performed simultaneously, resulting in the acquisition of multiple manifolds that share the same parameter space
For example, sensor networks --- such as camera networks or microphone arrays --- typically observe a single event from a variety of vantage points, while the underlying phenomenon can often be described by a set of common global parameters (such as the location and orientation of the objects of interest)
Similarly, when sensing a single phenomenon using multiple modalities, such as video and audio, the underlying phenomenon may again be described by a single parameterization that spans all modalities
In such cases, we will show that it is advantageous to model this joint structure contained in the ensemble of manifolds as opposed to simply treating each manifold independently
Thus we introduce the concept of the  joint manifold : a model for the concatenation of the data vectors observed by the group of sensors
Joint manifolds enable the development of improved manifold-based learning and estimation algorithms that exploit this structure
Furthermore, they can be applied to data of any modality and dimensionality
In this work we conduct a careful examination of the theoretical properties of joint manifolds
In particular, we compare joint manifolds to their component manifolds to see how quantities like geodesic distances, curvature, branch separation, and condition number are affected
We then observe that these properties lead to improved performance and noise-tolerance for a variety of signal processing algorithms when they exploit the joint manifold structure, as opposed to processing data from each manifold separately
We also illustrate how this joint manifold structure can be exploited through a simple and efficient data fusion algorithm that uses random projections, which can also be applied to multimodal data
Related prior work has studied  manifold alignment , where the goal is to discover maps between several datasets that are governed by the same underlying low-dimensional structure
Lafon et al \ proposed an algorithm to obtain a one-to-one matching between data points from several manifold-modeled classes~ CITATION
The algorithm first applies dimensionality reduction using diffusion maps to obtain data representations that encode the intrinsic geometry of the class
Then, an affine function that matches a set of landmark points is computed and applied to the remainder of the datasets
This concept was extended by Wang and Mahadevan, who apply Procrustes analysis on the dimensionality-reduced datasets to obtain an alignment function between a pair of manifolds~ CITATION
Since an alignment function is provided instead of a data point matching, the mapping obtained is applicable for the entire manifold rather than for the set of sampled points
In our setting, we assume that either ( i ) the manifold alignment is provided intrinsically via synchronization between the different sensors or ( ii ) the manifolds have been aligned using one of the approaches described above
Our main focus is a theoretical analysis of the benefits provided by analyzing the joint manifold versus solving our task of interest separately on each of the manifolds observed by individual sensors
This paper is organized as follows
Section~ introduces and establishes some basic properties of joint manifolds
Section~ considers the application of joint manifolds to the tasks of classification and manifold learning
Section~ then describes an efficient method for processing and aggregating data when it lies on a joint manifold, and Section~ concludes with discussion
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tex                                                                                       0000644 0000000 0000000 00000014040 11307647375 012166  0                                                                                                    ustar   root                            root                                                                                                                                                                                                                   \documentclass[12pt]{article} \input{preamble}  \title{A Theoretical Analysis of Joint Manifolds}  Mark A
Davenport, Chinmay Hegde, Marco F
Duarte, \\ and Richard G
Baraniuk \protect\\\protect\\ Rice University \protect\\ Department of Electrical and Computer Engineering\protect\\ Technical Report TREE0901}     \end{document}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 jam
