Heterogeneous datasets representation and learning using diffusion maps and Laplacian pyramids

Abstract

The diffusion maps together with the geometric harmonics provide a method for describing the geometry of high dimensional data and for extending these descriptions to new data points and to functions, which are defined on the data. This method suffers from two limitations. First, even though real-life data is often heterogeneous , the assumption in diffusion maps is that the attributes of the processed dataset are comparable. Second, application of the geometric harmonics requires careful setting for the correct extension scale and condition number. In this paper, we propose a method for representing and learning heterogeneous datasets by using diffusion maps for unifying and embedding heterogeneous dataset and by replacing the geometric harmonics with the Laplacian pyramid extension. Experimental results on three benchmark datasets demonstrate how the learning process becomes straightforward when the constructed representation smoothly parameterizes the task-related function.

A Side Scan Sonar Image Target Detection Algorithm Based on a Neutrosophic Set and Diffusion Maps

“…If the CPU is insufficiently strong, computing the DM for the entire dataset X is difficult. Therefore, to solve the problem, a DM can be constructed by using random samples of X and then by extending it to all points using the Nyström extension method [9,20].…”

Diffusion Maps for dimensionality reduction and visualization of meteorological data

“…From a general point of view LP is a multiscale algorithm for extending sample-based function values (in our case, the eigenvectors ψ j (x i )) that uses different scalings for different resolutions. More precisely, we can apply them [16] to approximate a function f from its values f (x k ) on a sample S = {x 1 , . .…”

“…The first one is an extension to the non-symmetric transition matrix P of the classical Nyström formula [15] for symmetric, positive semidefinite matrices derived from a kernel that extends the eigenvectors of the sample kernel matrix to the eigenfunctions of the underlying integral operator. The second one, the Laplacian Pyramids algorithm [16], also relies on a kernel representation but starts from the discrete sample values f (x i ) (in our case, the eigenvectors of the sample based Markov transition matrix P ) of a certain function f (in our case, the general eigenfunctions), and seeks a multiscale representation of f that allows to approximate the values f (x) from an appropriate multiscale combination of the sample values f (x i ).…”

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“…In [10] a method is proposed to adapt DM to work with heterogeneous features just by dealing separately with groups of attributes that are deemed to be homogeneous. More …”

“…Moreover, a main drawback (as it also happens in spectral DR) is the difficulty to apply the computed DM projection to new, unseen patterns. There are several proposals for this such as Nyström formulae [4] or Laplacian Pyramids [10], but this is still an area where further work is needed.…”