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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.21107">arXiv:2410.21107</a> <span> [<a href="https://arxiv.org/pdf/2410.21107">pdf</a>, <a href="https://arxiv.org/format/2410.21107">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Lin%2C+Y+E">Ya-Wei Eileen Lin</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Mishne%2C+G">Gal Mishne</a>, <a href="/search/cs?searchtype=author&query=Talmon%2C+R">Ronen Talmon</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.21107v2-abstract-short" style="display: inline;"> 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 hyper… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21107v2-abstract-full').style.display = 'inline'; document.getElementById('2410.21107v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.21107v2-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21107v2-abstract-full').style.display = 'none'; document.getElementById('2410.21107v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.06812">arXiv:2406.06812</a> <span> [<a href="https://arxiv.org/pdf/2406.06812">pdf</a>, <a href="https://arxiv.org/format/2406.06812">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> On Learning what to Learn: heterogeneous observations of dynamics and establishing (possibly causal) relations among them </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Sroczynski%2C+D+W">David W. Sroczynski</a>, <a href="/search/cs?searchtype=author&query=Dietrich%2C+F">Felix Dietrich</a>, <a href="/search/cs?searchtype=author&query=Koronaki%2C+E+D">Eleni D. Koronaki</a>, <a href="/search/cs?searchtype=author&query=Talmon%2C+R">Ronen Talmon</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Bollt%2C+E">Erik Bollt</a>, <a href="/search/cs?searchtype=author&query=Kevrekidis%2C+I+G">Ioannis G. Kevrekidis</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.06812v1-abstract-short" style="display: inline;"> Before we attempt to learn a function between two (sets of) observables of a physical process, we must first decide what the inputs and what the outputs of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of initially deciding ``the right quantities'' to relate through such a function, and then proceed to learn it. This is accomplished by processing multiple… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.06812v1-abstract-full').style.display = 'inline'; document.getElementById('2406.06812v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.06812v1-abstract-full" style="display: none;"> Before we attempt to learn a function between two (sets of) observables of a physical process, we must first decide what the inputs and what the outputs of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of initially deciding ``the right quantities'' to relate through such a function, and then proceed to learn it. This is accomplished by processing multiple simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: multiple observation processes of the system. We thus determine (a) what subsets of observables are common between the observation processes (and therefore observable from each other, relatable through a function); and (b) what information is unrelated to these common observables, and therefore particular to each observation process, and not contributing to the desired function. Any data-driven function approximation technique can subsequently be used to learn the input-output relation, from k-nearest neighbors and Geometric Harmonics to Gaussian Processes and Neural Networks. Two particular ``twists'' of the approach are discussed. The first has to do with the identifiability of particular quantities of interest from the measurements. We now construct mappings from a single set of observations of one process to entire level sets of measurements of the process, consistent with this single set. The second attempts to relate our framework to a form of causality: if one of the observation processes measures ``now'', while the second observation process measures ``in the future'', the function to be learned among what is common across observation processes constitutes a dynamical model for the system evolution. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.06812v1-abstract-full').style.display = 'none'; document.getElementById('2406.06812v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.13155">arXiv:2312.13155</a> <span> [<a href="https://arxiv.org/pdf/2312.13155">pdf</a>, <a href="https://arxiv.org/format/2312.13155">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Gappy local conformal auto-encoders for heterogeneous data fusion: in praise of rigidity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Peterfreund%2C+E">Erez Peterfreund</a>, <a href="/search/cs?searchtype=author&query=Burak%2C+I">Iryna Burak</a>, <a href="/search/cs?searchtype=author&query=Lindenbaum%2C+O">Ofir Lindenbaum</a>, <a href="/search/cs?searchtype=author&query=Gimlett%2C+J">Jim Gimlett</a>, <a href="/search/cs?searchtype=author&query=Dietrich%2C+F">Felix Dietrich</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Kevrekidis%2C+I+G">Ioannis G. Kevrekidis</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.13155v1-abstract-short" style="display: inline;"> Fusing measurements from multiple, heterogeneous, partial sources, observing a common object or process, poses challenges due to the increasing availability of numbers and types of sensors. In this work we propose, implement and validate an end-to-end computational pipeline in the form of a multiple-auto-encoder neural network architecture for this task. The inputs to the pipeline are several sets… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13155v1-abstract-full').style.display = 'inline'; document.getElementById('2312.13155v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.13155v1-abstract-full" style="display: none;"> Fusing measurements from multiple, heterogeneous, partial sources, observing a common object or process, poses challenges due to the increasing availability of numbers and types of sensors. In this work we propose, implement and validate an end-to-end computational pipeline in the form of a multiple-auto-encoder neural network architecture for this task. The inputs to the pipeline are several sets of partial observations, and the result is a globally consistent latent space, harmonizing (rigidifying, fusing) all measurements. The key enabler is the availability of multiple slightly perturbed measurements of each instance:, local measurement, "bursts", that allows us to estimate the local distortion induced by each instrument. We demonstrate the approach in a sequence of examples, starting with simple two-dimensional data sets and proceeding to a Wi-Fi localization problem and to the solution of a "dynamical puzzle" arising in spatio-temporal observations of the solutions of Partial Differential Equations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13155v1-abstract-full').style.display = 'none'; document.getElementById('2312.13155v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.18962">arXiv:2305.18962</a> <span> [<a href="https://arxiv.org/pdf/2305.18962">pdf</a>, <a href="https://arxiv.org/format/2305.18962">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Lin%2C+Y+E">Ya-Wei Eileen Lin</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Mishne%2C+G">Gal Mishne</a>, <a href="/search/cs?searchtype=author&query=Talmon%2C+R">Ronen Talmon</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.18962v1-abstract-short" style="display: inline;"> 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… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.18962v1-abstract-full').style.display = 'inline'; document.getElementById('2305.18962v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.18962v1-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.18962v1-abstract-full').style.display = 'none'; document.getElementById('2305.18962v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.14747">arXiv:2107.14747</a> <span> [<a href="https://arxiv.org/pdf/2107.14747">pdf</a>, <a href="https://arxiv.org/format/2107.14747">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s10208-022-09558-8">10.1007/s10208-022-09558-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A common variable minimax theorem for graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Marshall%2C+N+F">Nicholas F. Marshall</a>, <a href="/search/cs?searchtype=author&query=Steinerberger%2C+S">Stefan Steinerberger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2107.14747v1-abstract-short" style="display: inline;"> Let $\mathcal{G} = \{G_1 = (V, E_1), \dots, G_m = (V, E_m)\}$ be a collection of $m$ graphs defined on a common set of vertices $V$ but with different edge sets $E_1, \dots, E_m$. Informally, a function $f :V \rightarrow \mathbb{R}$ is smooth with respect to $G_k = (V,E_k)$ if $f(u) \sim f(v)$ whenever $(u, v) \in E_k$. We study the problem of understanding whether there exists a nonconstant funct… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.14747v1-abstract-full').style.display = 'inline'; document.getElementById('2107.14747v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.14747v1-abstract-full" style="display: none;"> Let $\mathcal{G} = \{G_1 = (V, E_1), \dots, G_m = (V, E_m)\}$ be a collection of $m$ graphs defined on a common set of vertices $V$ but with different edge sets $E_1, \dots, E_m$. Informally, a function $f :V \rightarrow \mathbb{R}$ is smooth with respect to $G_k = (V,E_k)$ if $f(u) \sim f(v)$ whenever $(u, v) \in E_k$. We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in $\mathcal{G}$, simultaneously, and how to find it if it exists. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.14747v1-abstract-full').style.display = 'none'; document.getElementById('2107.14747v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, 11 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.00402">arXiv:2006.00402</a> <span> [<a href="https://arxiv.org/pdf/2006.00402">pdf</a>, <a href="https://arxiv.org/ps/2006.00402">ps</a>, <a href="https://arxiv.org/format/2006.00402">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Doubly-Stochastic Normalization of the Gaussian Kernel is Robust to Heteroskedastic Noise </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Landa%2C+B">Boris Landa</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Kluger%2C+Y">Yuval Kluger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.00402v2-abstract-short" style="display: inline;"> A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When the data points reside in Euclidean space, a widespread approach is to from an affinity matrix by the Gaussian kernel with pairwise distances, and to follow with a certain normalization (e.g. the row-stochastic normalization or its symmetric variant). We d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.00402v2-abstract-full').style.display = 'inline'; document.getElementById('2006.00402v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.00402v2-abstract-full" style="display: none;"> A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When the data points reside in Euclidean space, a widespread approach is to from an affinity matrix by the Gaussian kernel with pairwise distances, and to follow with a certain normalization (e.g. the row-stochastic normalization or its symmetric variant). We demonstrate that the doubly-stochastic normalization of the Gaussian kernel with zero main diagonal (i.e., no self loops) is robust to heteroskedastic noise. That is, the doubly-stochastic normalization is advantageous in that it automatically accounts for observations with different noise variances. Specifically, we prove that in a suitable high-dimensional setting where heteroskedastic noise does not concentrate too much in any particular direction in space, the resulting (doubly-stochastic) noisy affinity matrix converges to its clean counterpart with rate $m^{-1/2}$, where $m$ is the ambient dimension. We demonstrate this result numerically, and show that in contrast, the popular row-stochastic and symmetric normalizations behave unfavorably under heteroskedastic noise. Furthermore, we provide examples of simulated and experimental single-cell RNA sequence data with intrinsic heteroskedasticity, where the advantage of the doubly-stochastic normalization for exploratory analysis is evident. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.00402v2-abstract-full').style.display = 'none'; document.getElementById('2006.00402v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2004.07234">arXiv:2004.07234</a> <span> [<a href="https://arxiv.org/pdf/2004.07234">pdf</a>, <a href="https://arxiv.org/format/2004.07234">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1073/pnas.2014627117">10.1073/pnas.2014627117 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> LOCA: LOcal Conformal Autoencoder for standardized data coordinates </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Peterfreund%2C+E">Erez Peterfreund</a>, <a href="/search/cs?searchtype=author&query=Lindenbaum%2C+O">Ofir Lindenbaum</a>, <a href="/search/cs?searchtype=author&query=Dietrich%2C+F">Felix Dietrich</a>, <a href="/search/cs?searchtype=author&query=Bertalan%2C+T">Tom Bertalan</a>, <a href="/search/cs?searchtype=author&query=Gavish%2C+M">Matan Gavish</a>, <a href="/search/cs?searchtype=author&query=Kevrekidis%2C+I+G">Ioannis G. Kevrekidis</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2004.07234v2-abstract-short" style="display: inline;"> We propose a deep-learning based method for obtaining standardized data coordinates from scientific measurements.Data observations are modeled as samples from an unknown, non-linear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized latent variables. By leveraging a repeated measurement sampling strategy, we present a method for learning an embedding in… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.07234v2-abstract-full').style.display = 'inline'; document.getElementById('2004.07234v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2004.07234v2-abstract-full" style="display: none;"> We propose a deep-learning based method for obtaining standardized data coordinates from scientific measurements.Data observations are modeled as samples from an unknown, non-linear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized latent variables. By leveraging a repeated measurement sampling strategy, we present a method for learning an embedding in $\mathbb{R}^d$ that is isometric to the latent variables of the manifold. These data coordinates, being invariant under smooth changes of variables, enable matching between different instrumental observations of the same phenomenon. Our embedding is obtained using a LOcal Conformal Autoencoder (LOCA), an algorithm that constructs an embedding to rectify deformations by using a local z-scoring procedure while preserving relevant geometric information. We demonstrate the isometric embedding properties of LOCA on various model settings and observe that it exhibits promising interpolation and extrapolation capabilities. Finally, we apply LOCA to single-site Wi-Fi localization data, and to $3$-dimensional curved surface estimation based on a $2$-dimensional projection. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.07234v2-abstract-full').style.display = 'none'; document.getElementById('2004.07234v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.06803">arXiv:1810.06803</a> <span> [<a href="https://arxiv.org/pdf/1810.06803">pdf</a>, <a href="https://arxiv.org/format/1810.06803">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Co-manifold learning with missing data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Mishne%2C+G">Gal Mishne</a>, <a href="/search/cs?searchtype=author&query=Chi%2C+E+C">Eric C. Chi</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.06803v1-abstract-short" style="display: inline;"> 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 set… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.06803v1-abstract-full').style.display = 'inline'; document.getElementById('1810.06803v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.06803v1-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.06803v1-abstract-full').style.display = 'none'; document.getElementById('1810.06803v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, 9 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.05006">arXiv:1709.05006</a> <span> [<a href="https://arxiv.org/pdf/1709.05006">pdf</a>, <a href="https://arxiv.org/format/1709.05006">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> </div> </div> <p class="title is-5 mathjax"> Two-sample Statistics Based on Anisotropic Kernels </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Cheng%2C+X">Xiuyuan Cheng</a>, <a href="/search/cs?searchtype=author&query=Cloninger%2C+A">Alexander Cloninger</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.05006v3-abstract-short" style="display: inline;"> The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kerne… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.05006v3-abstract-full').style.display = 'inline'; document.getElementById('1709.05006v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.05006v3-abstract-full" style="display: none;"> The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between $n$ data points and a set of $n_R$ reference points, where $n_R$ can be drastically smaller than $n$. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as $\|p-q\| \sqrt{n} \to \infty $, and a finite-sample lower bound of the testing power is obtained. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.05006v3-abstract-full').style.display = 'none'; document.getElementById('1709.05006v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 August, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1708.05768">arXiv:1708.05768</a> <span> [<a href="https://arxiv.org/pdf/1708.05768">pdf</a>, <a href="https://arxiv.org/format/1708.05768">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantitative Methods">q-bio.QM</span> </div> </div> <p class="title is-5 mathjax"> Data-Driven Tree Transforms and Metrics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Mishne%2C+G">Gal Mishne</a>, <a href="/search/cs?searchtype=author&query=Talmon%2C+R">Ronen Talmon</a>, <a href="/search/cs?searchtype=author&query=Cohen%2C+I">Israel Cohen</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Kluger%2C+Y">Yuval Kluger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1708.05768v1-abstract-short" style="display: inline;"> We consider the analysis of high dimensional data given in the form of a matrix with columns consisting of observations and rows consisting of features. Often the data is such that the observations do not reside on a regular grid, and the given order of the features is arbitrary and does not convey a notion of locality. Therefore, traditional transforms and metrics cannot be used for data organiza… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.05768v1-abstract-full').style.display = 'inline'; document.getElementById('1708.05768v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1708.05768v1-abstract-full" style="display: none;"> We consider the analysis of high dimensional data given in the form of a matrix with columns consisting of observations and rows consisting of features. Often the data is such that the observations do not reside on a regular grid, and the given order of the features is arbitrary and does not convey a notion of locality. Therefore, traditional transforms and metrics cannot be used for data organization and analysis. In this paper, our goal is to organize the data by defining an appropriate representation and metric such that they respect the smoothness and structure underlying the data. We also aim to generalize the joint clustering of observations and features in the case the data does not fall into clear disjoint groups. For this purpose, we propose multiscale data-driven transforms and metrics based on trees. Their construction is implemented in an iterative refinement procedure that exploits the co-dependencies between features and observations. Beyond the organization of a single dataset, our approach enables us to transfer the organization learned from one dataset to another and to integrate several datasets together. We present an application to breast cancer gene expression analysis: learning metrics on the genes to cluster the tumor samples into cancer sub-types and validating the joint organization of both the genes and the samples. We demonstrate that using our approach to combine information from multiple gene expression cohorts, acquired by different profiling technologies, improves the clustering of tumor samples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.05768v1-abstract-full').style.display = 'none'; document.getElementById('1708.05768v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 August, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, 5 figures. Accepted to IEEE Transactions on Signal and Information Processing over Networks</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.03195">arXiv:1612.03195</a> <span> [<a href="https://arxiv.org/pdf/1612.03195">pdf</a>, <a href="https://arxiv.org/format/1612.03195">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Pattern Formation and Solitons">nlin.PS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Geometry">cs.CG</span> </div> </div> <p class="title is-5 mathjax"> No equations, no parameters, no variables: data, and the reconstruction of normal forms by learning informed observation geometries </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Yair%2C+O">Or Yair</a>, <a href="/search/cs?searchtype=author&query=Talmon%2C+R">Ronen Talmon</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Kevrekidis%2C+I+G">Ioannis G. Kevrekidis</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1612.03195v1-abstract-short" style="display: inline;"> The discovery of physical laws consistent with empirical observations lies at the heart of (applied) science and engineering. These laws typically take the form of nonlinear differential equations depending on parameters, dynamical systems theory provides, through the appropriate normal forms, an "intrinsic", prototypical characterization of the types of dynamical regimes accessible to a given mod… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.03195v1-abstract-full').style.display = 'inline'; document.getElementById('1612.03195v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.03195v1-abstract-full" style="display: none;"> The discovery of physical laws consistent with empirical observations lies at the heart of (applied) science and engineering. These laws typically take the form of nonlinear differential equations depending on parameters, dynamical systems theory provides, through the appropriate normal forms, an "intrinsic", prototypical characterization of the types of dynamical regimes accessible to a given model. Using an implementation of data-informed geometry learning we directly reconstruct the relevant "normal forms": a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics. Interestingly, the state variables and the parameters of these realizations are inferred from the empirical observations, without prior knowledge or understanding, they parametrize the dynamics {\em intrinsically}, without explicit reference to fundamental physical quantities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.03195v1-abstract-full').style.display = 'none'; document.getElementById('1612.03195v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 November, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1509.07385">arXiv:1509.07385</a> <span> [<a href="https://arxiv.org/pdf/1509.07385">pdf</a>, <a href="https://arxiv.org/format/1509.07385">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Neural and Evolutionary Computing">cs.NE</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.acha.2016.04.003">10.1016/j.acha.2016.04.003 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Provable approximation properties for deep neural networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Shaham%2C+U">Uri Shaham</a>, <a href="/search/cs?searchtype=author&query=Cloninger%2C+A">Alexander Cloninger</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1509.07385v3-abstract-short" style="display: inline;"> We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $螕\subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating $f$. The size of the network depends on dimension and curvature of the manifold $螕$, the complexity of $f$, in terms of its wavelet description, and only weakly on the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.07385v3-abstract-full').style.display = 'inline'; document.getElementById('1509.07385v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1509.07385v3-abstract-full" style="display: none;"> We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $螕\subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating $f$. The size of the network depends on dimension and curvature of the manifold $螕$, the complexity of $f$, in terms of its wavelet description, and only weakly on the ambient dimension $m$. Essentially, our network computes wavelet functions, which are computed from Rectified Linear Units (ReLU) <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.07385v3-abstract-full').style.display = 'none'; document.getElementById('1509.07385v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">accepted for publication in Applied and Computational Harmonic Analysis</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1507.00220">arXiv:1507.00220</a> <span> [<a href="https://arxiv.org/pdf/1507.00220">pdf</a>, <a href="https://arxiv.org/format/1507.00220">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Bigeometric Organization of Deep Nets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Cloninger%2C+A">Alexander Cloninger</a>, <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Downing%2C+N">Nicholas Downing</a>, <a href="/search/cs?searchtype=author&query=Krumholz%2C+H+M">Harlan M. Krumholz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1507.00220v1-abstract-short" style="display: inline;"> In this paper, we build an organization of high-dimensional datasets that cannot be cleanly embedded into a low-dimensional representation due to missing entries and a subset of the features being irrelevant to modeling functions of interest. Our algorithm begins by defining coarse neighborhoods of the points and defining an expected empirical function value on these neighborhoods. We then generat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1507.00220v1-abstract-full').style.display = 'inline'; document.getElementById('1507.00220v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1507.00220v1-abstract-full" style="display: none;"> In this paper, we build an organization of high-dimensional datasets that cannot be cleanly embedded into a low-dimensional representation due to missing entries and a subset of the features being irrelevant to modeling functions of interest. Our algorithm begins by defining coarse neighborhoods of the points and defining an expected empirical function value on these neighborhoods. We then generate new non-linear features with deep net representations tuned to model the approximate function, and re-organize the geometry of the points with respect to the new representation. Finally, the points are locally z-scored to create an intrinsic geometric organization which is independent of the parameters of the deep net, a geometry designed to assure smoothness with respect to the empirical function. We examine this approach on data from the Center for Medicare and Medicaid Services Hospital Quality Initiative, and generate an intrinsic low-dimensional organization of the hospitals that is smooth with respect to an expert driven function of quality. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1507.00220v1-abstract-full').style.display = 'none'; document.getElementById('1507.00220v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 July, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1209.0245">arXiv:1209.0245</a> <span> [<a href="https://arxiv.org/pdf/1209.0245">pdf</a>, <a href="https://arxiv.org/format/1209.0245">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.acha.2013.03.001">10.1016/j.acha.2013.03.001 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Diffusion maps for changing data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Hirn%2C+M+J">Matthew J. Hirn</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1209.0245v3-abstract-short" style="display: inline;"> Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph associated with it changes depending on some set of parameters. We analyze this type of data in terms of the diffusion distance and the corresponding diffusion map. As the data changes over the parame… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1209.0245v3-abstract-full').style.display = 'inline'; document.getElementById('1209.0245v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1209.0245v3-abstract-full" style="display: none;"> Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph associated with it changes depending on some set of parameters. We analyze this type of data in terms of the diffusion distance and the corresponding diffusion map. As the data changes over the parameter space, the low dimensional embedding changes as well. We give a way to go between these embeddings, and furthermore, map them all into a common space, allowing one to track the evolution of X in its intrinsic geometry. A global diffusion distance is also defined, which gives a measure of the global behavior of the data over the parameter space. Approximation theorems in terms of randomly sampled data are presented, as are potential applications. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1209.0245v3-abstract-full').style.display = 'none'; document.getElementById('1209.0245v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 July, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 September, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">38 pages. 9 figures. To appear in Applied and Computational Harmonic Analysis. v2: Several minor changes beyond just typos. v3: Minor typo corrected, added DOI</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Applied and Computational Harmonic Analysis, Volume 36, Issue 1, January 2014, Pages 79-107 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1209.0237">arXiv:1209.0237</a> <span> [<a href="https://arxiv.org/pdf/1209.0237">pdf</a>, <a href="https://arxiv.org/ps/1209.0237">ps</a>, <a href="https://arxiv.org/format/1209.0237">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.acha.2013.01.001">10.1016/j.acha.2013.01.001 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Bi-stochastic kernels via asymmetric affinity functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Coifman%2C+R+R">Ronald R. Coifman</a>, <a href="/search/cs?searchtype=author&query=Hirn%2C+M+J">Matthew J. Hirn</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1209.0237v4-abstract-short" style="display: inline;"> In this short letter we present the construction of a bi-stochastic kernel p for an arbitrary data set X that is derived from an asymmetric affinity function 伪. The affinity function 伪 measures the similarity between points in X and some reference set Y. Unlike other methods that construct bi-stochastic kernels via some convergent iteration process or through solving an optimization problem, the c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1209.0237v4-abstract-full').style.display = 'inline'; document.getElementById('1209.0237v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1209.0237v4-abstract-full" style="display: none;"> In this short letter we present the construction of a bi-stochastic kernel p for an arbitrary data set X that is derived from an asymmetric affinity function 伪. The affinity function 伪 measures the similarity between points in X and some reference set Y. Unlike other methods that construct bi-stochastic kernels via some convergent iteration process or through solving an optimization problem, the construction presented here is quite simple. Furthermore, it can be viewed through the lens of out of sample extensions, making it useful for massive data sets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1209.0237v4-abstract-full').style.display = 'none'; document.getElementById('1209.0237v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 July, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 September, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">5 pages. v2: Expanded upon the first paragraph of subsection 2.1. v3: Minor changes and edits. v4: Edited comments and added DOI</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Applied and Computational Harmonic Analysis, volume 35, number 1, pages 177-180, July, 2013 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 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