Finding meaningful distances between high-dimensional data samples is an important scientific task. To this end, we propose a new tree-Wasserstein distance (TWD) for high-dimensional data with two key aspects. First, our TWD is specifically designed for data with a latent feature hierarchy, i.e., the features lie in a hierarchical space, in contrast to the usual focus on embedding samples in hyperbolic space. Second, while the conventional use of TWD is to speed up the computation of the Wasserstein distance, we use its inherent tree as a means to learn the latent feature hierarchy. The key idea of our method is to embed the features into a multi-scale hyperbolic space using diffusion geometry and then present a new tree decoding method by establishing analogies between the hyperbolic embedding and trees. We show that our TWD computed based on data observations provably recovers the TWD defined with the latent feature hierarchy and that its computation is efficient and scalable. We showcase the usefulness of the proposed TWD in applications to word-document and single-cell RNA-sequencing datasets, demonstrating its advantages over existing TWDs and methods based on pre-trained models.

High-dimensional data often exhibit hierarchical structures in both modes: samples and features. Yet, most existing approaches for hierarchical representation learning consider only one mode at a time. In this work, we propose an unsupervised method for jointly learning hierarchical representations of samples and features via Tree-Wasserstein Distance (TWD). Our method alternates between the two data modes. It first constructs a tree for one mode, then computes a TWD for the other mode based on that tree, and finally uses the resulting TWD to build the second mode’s tree. By repeatedly alternating through these steps, the method gradually refines both trees and the corresponding TWDs, capturing meaningful hierarchical representations of the data. We provide a theoretical analysis showing that our method converges. We show that our method can be integrated into hyperbolic graph convolutional networks as a pre-processing technique, improving performance in link prediction and node classification tasks. In addition, our method outperforms baselines in sparse approximation and unsupervised Wasserstein distance learning tasks on word-document and single-cell RNA-sequencing datasets.

Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central approach to manifold learning, and hyperbolic geometry. Specifically, using diffusion geometry, we build multi-scale densities on the data, aimed to reveal their hierarchical structure, and then embed them into a product of hyperbolic spaces. We show theoretically that our embedding and distance recover the underlying hierarchical structure. In addition, we demonstrate the efficacy of the proposed method and its advantages compared to existing methods on graph embedding benchmarks and hierarchical datasets.

We propose a new fast method of measuring distances between large numbers of related high dimensional datasets called the Diffusion Earth Mover’s Distance (EMD). We model the datasets as distributions supported on common data graph that is derived from the affinity matrix computed on the combined data. In such cases where the graph is a discretization of an underlying Riemannian closed manifold, we prove that Diffusion EMD is topologically equivalent to the standard EMD with a geodesic ground distance. Diffusion EMD can be computed in {Õ}(n) time and is more accurate than similarly fast algorithms such as tree-based EMDs. We also show Diffusion EMD is fully differentiable, making it amenable to future uses in gradient-descent frameworks such as deep neural networks. Finally, we demonstrate an application of Diffusion EMD to single cell data collected from 210 COVID-19 patient samples at Yale New Haven Hospital. Here, Diffusion EMD can derive distances between patients on the manifold of cells at least two orders of magnitude faster than equally accurate methods. This distance matrix between patients can be embedded into a higher level patient manifold which uncovers structure and heterogeneity in patients. More generally, Diffusion EMD is applicable to all datasets that are massively collected in parallel in many medical and biological systems.

Representation learning is typically applied to only one mode of a data matrix, either its rows or columns. Yet in many applications, there is an underlying geometry to both the rows and the columns. We propose utilizing this coupled structure to perform co-manifold learning: uncovering the underlying geometry of both the rows and the columns of a given matrix, where we focus on a missing data setting. Our unsupervised approach consists of three components. We first solve a family of optimization problems to estimate a complete matrix at multiple scales of smoothness. We then use this collection of smooth matrix estimates to compute pairwise distances on the rows and columns based on a new multi-scale metric that implicitly introduces a coupling between the rows and the columns. Finally, we construct row and column representations from these multi-scale metrics. We demonstrate that our approach outperforms competing methods in both data visualization and clustering.

In this paper, we address the problem of multimodal signal processing and present a manifold learning method to extract the common source of variability from multiple measurements. This method is based on alternating-diffusion and is particularly adapted to time series. We show that the common source of variability is extracted from multiple sensors as if it were the only source of variability, extracted by a standard manifold learning method from a single sensor, without the influence of the sensor-specific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods.

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.

In this paper, we present a supervised graph-based framework for sequential processing and employ it to the problem of transient interference suppression. Transients typically consist of an initial peak followed by decaying short-duration oscillations. Such sounds, e.g., keyboard typing and door knocking, often arise as an interference in everyday applications: hearing aids, hands-free accessories, mobile phones, and conference-room devices. We describe a graph construction using a noisy speech signal and training recordings of typical transients. The main idea is to capture the transient interference structure, which may emerge from the construction of the graph. The graph parametrization is then viewed as a data-driven model of the transients and utilized to define a filter that extracts the transients from noisy speech measurements. Unlike previous transient interference suppression studies, in this work the graph is constructed in advance from training recordings. Then, the graph is extended to newly acquired measurements, providing a sequential filtering framework of noisy speech.

We introduce a graph Laplacian-based algorithm for the tomographic reconstruction of a planar object from its projections taken at random unknown directions. A Laplace-type operator is constructed on the data set of projections, and the eigenvectors of this operator reveal the projection orientations. The algorithm is shown to successfully reconstruct the Shepp-Logan phantom from its noisy projections. Such a reconstruction algorithm is desirable for the structuring of certain biological proteins using cryo-electron microscopy.

We describe methods for the classification of brain state using quantitative analysis of the EEG (QEEG). Neurometric analysis of EEG collected from the 19 standard locations of the International 10-20 System already provides such a tool. In this work we demonstrate the effectiveness of this approach when the available inputs are reduced to a set of five frontal electrodes. This system has applications in certain critical clinical care situations, such as emergency room triage, when a full EEG might be unavailable, inconvenient, or time-consuming. Additionally, we augment the standard neurometric QEEG analysis with local discriminant basis features of the power spectrum and microstate-like features which exploit the rich temporal structure of the EEG. These enhancements provide clear gains in sensitivity and specificity on a representative database.

Data fusion and multicue data matching are fundamental tasks of high-dimensional data analysis. In this paper, we apply the recently introduced diffusion framework to address these tasks. Our contribution is three-fold: First, we present the Laplace-Beltrami approach for computing density invariant embeddings which are essential for integrating different sources of data. Second, we describe a refinement of the Nyström extension algorithm called "geometric harmonics." We also explain how to use this tool for data assimilation. Finally, we introduce a multicue data matching scheme based on nonlinear spectral graphs alignment. The effectiveness of the presented schemes is validated by applying it to the problems of lipreading and image sequence alignment.

This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the data by the first few eigenvectors of the corresponding Markov matrix is optimal under a certain mean squared error criterion. Furthermore, assuming that data points are random samples from a density p(x) = e-U (x) we identify these eigenvectors as discrete approximations of eigenfunctions of a Fokker-Planck operator in a potential 2U (x) with reflecting boundary conditions. Finally, applying known results regarding the eigenvalues and eigenfunctions of the continuous Fokker-Planck operator, we provide a mathematical justification for the success of spectral clustering and dimensional reduction algorithms based on these first few eigenvectors. This analysis elucidates, in terms of the characteristics of diffusion processes, many empirical findings regarding spectral clustering algorithms. Keywords: Algorithms and architectures, learning theory.

Image coding methods based on adaptive wavelet transforms and those employing zerotree quantization have been shown to be successful. We present a general zerotree structure for an arbitrary wavelet packet geometry in an image coding framework. A fast basis selection algorithm is developed; it uses a Markov chain based cost estimate of encoding the image using this structure. As a result, our adaptive wavelet zerotree image coder has a relatively low computational complexity, performs comparably to state-of-the-art image coders, and is capable of progressively encoding images.

The main contribution of this work is a new paradigm for image representation and image compression. We describe a new multilayered representation technique for images. An image is parsed into a superposition of coherent layers: piecewise smooth regions layer, textures layer, etc. The multilayered decomposition algorithm consists in a cascade of compressions applied successively to the image itself and to the residuals that resulted from the previous compressions. During each iteration of the algorithm, we code the residual part in a lossy way: we only retain the most significant structures of the residual part, which results in a sparse representation. Each layer is encoded independently with a different transform, or basis, at a different bitrate, and the combination of the compressed layers can always be reconstructed in a meaningful way. The strength of the multilayer approach comes from the fact that different sets of basis functions complement each others: some of the basis functions will give reasonable account of the large trend of the data, while others will catch the local transients, or the oscillatory patterns. This multilayered representation has a lot of beautiful applications in image understanding, and image and video coding. We have implemented the algorithm and we have studied its capabilities.

In this paper, work on a new wavelet packet basis selection paradigm is reported which emphasizes the crucial role of the quantization strategy being used. This paradigm is coupled with a new Markov chain based estimation of the cost of zerotree quantization to develop a progressive wavelet packet image coder which gives better results than its wavelet counterpart.

Compression is a relatively new introduced technique for seismic data operations. The main drive behind the use of data compression in seismic data is the very large size of seismic data acquired. Some of the most recent acquired marine seismic data sets exceed 10 Tbytes, and in fact there are currently seismic surveys planned with a volume of around 120 Tbytes. Thus, the need to compress these very large seismic data files is imperative. Nevertheless, seismic data are quite different from the typical images used in image processing and multimedia applications. Some of their major differences are the data dynamic range exceeding 100 dB in theory, very often it is data with extensive oscillatory nature, the x and y directions represent different physical meaning, and there is significant amount of coherent noise which is often present in seismic data. Up to now some of the algorithms used for seismic data compression were based on some form of wavelet or local cosine transform, while using a uniform or quasiuniform quantization scheme and they finally employ a Huffman coding scheme. Using this family of compression algorithms we achieve compression results which are acceptable to geophysicists, only at low to moderate compression ratios. For higher compression ratios or higher decibel quality, significant compression artifacts are introduced in the reconstructed images, even with high-dimensional transforms. The objective of this paper is to achieve higher compression ratio, than achieved with the wavelet/uniform quantization/Huffman coding family of compression schemes, with a comparable level of residual noise. The goal is to achieve above 40 dB in the decompressed seismic data sets. Several established compression algorithms are reviewed, and some new compression algorithms are introduced. All of these compression techniques are applied to a good representation of seismic data sets, and their results are documented in this paper. One of the conclusions is that adaptive multiscale local cosine transform with different windows sizes performs well on all the seismic data sets and outperforms the other methods from the SNR point of view. All the described methods cover wide range of different data sets. Each data set will have his own best performed method chosen from this collection. The results were performed on four different seismic data sets. Special emphasis was given to achieve faster processing speed which is another critical issue that is examined in the paper. Some of these algorithms are also suitable for multimedia type compression.

Wavelet packets are an effective representation tool for adaptive waveform analysis of a given signal. We first combine the wavelet packet representation with zerotree quantization for image coding. A general zerotree structure is defined which can adapt itself to any arbitrary wavelet packet basis. We then describe an efficient coding algorithm based on this structure. Finally, the hypothesis for prediction of coefficients from coarser scale to finer scale is tested and its effectiveness is compared with that of zerotree hypothesis for wavelet coefficients.

Summary form only given. Presents a new fast wavelet packet compression algorithm that encodes very efficiently textured images. This fast wavelet packet compression technique relies on four stages: 1. Very fast convolution and decimation of the image with factorized filters. 2. Selection of a best basis in a large library of waveforms. The best basis is that basis which is best adapted to the content of the image. 3. Scanning of the wavelet packet coefficients by increasing frequency. This organization yields sequences of coefficients with a rapid decay. 4. Successive embedded approximation quantization, and entropy coding of the coefficients. We implemented the wavelet packet coder and decoder, and actual bit streams were created for each experiment. Our implementation used the 7-9 biorthogonal filters. We present the results of the algorithm, using the test image 512 x 512 Barbara. In order to evaluate the performance of the algorithm, we compared our algorithm to the SPIHT wavelet coder of Said and Pearlman (1996).

We describe a new multi-layered representation technique for images. An image is encoded as the superposition of one main approximation, and a sequence of residuals. The strength of the multi-layered method comes from the fact that we use different bases to encode the main approximation and the residuals. The different bases complement each others: some of the basis functions will give reasonable account of the large trend of the data, while others will catch the local transients, or the oscillatory patterns. By selecting different bases, we allow different features to be discovered in the image.

We construct a new algorithm for motion compensation of wavelet coefficients. The new approach outperforms standard block matching techniques both in terms of higher PSNR, and better perceptual quality. The algorithm works directly in the wavelet coefficients domain and has been successfully used for wavelet based very low bit rate video coding.

We construct a new adaptive basis that provides precise frequency localization and good spatial localization. We develop a compression algorithm that exploits this basis to obtain the most economical representation of an image in terms of textured patterns with different orientations, frequencies, sizes, and positions. The technique directly works in the Fourier domain and has potential applications for compression of richly textured images.

We describe extensions to the "best-basis" method which select orthonormal bases suitable for signal classification and regression problems from a large collection of orthonormal bases. For classification problems, we select the basis which maximizes relative entropy of time-frequency energy distributions among classes. For regression problems, we select the basis which tries to minimize the regression error. Once these bases are selected, a small number of most significant coordinates are fed into a traditional classifier or regression method such as linear discriminant analysis (LDA) or classification and regression tree (CART). The performance of these statistical methods is enhanced since the proposed methods reduce the dimensionality of the problems without losing important information for the problem at hand. The basis functions which are well-localized in the time-frequency plane are used as feature extractors. We also compare their performance with the traditional methods using a synthetic example.

Adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals. It permits efficient compression of a variety of signals, such as sound and images. The predefined libraries of modulated waveforms include orthogonal wavelet-packets and localized trigonometric functions, and have reasonably well-controlled time-frequency localization properties. The idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost. The method relies heavily on the remarkable orthogonality properties of the new libraries: all expansions in a given library conserve energy and are thus comparable. Several cost functionals are useful; one of the most attractive is Shannon entropy, which has a geometric interpretation in this context.>