Selecting data for training machine learning models is crucial since large, web-scraped, real datasets contain noisy artifacts that affect the quality and relevance of individual data points. These noisy artifacts will impact model performance. We formulate this problem as a data valuation task, assigning a value to data points in the training set according to how similar or dissimilar they are to a clean and curated validation set. Recently, *LAVA* (Just et al., 2023) demonstrated the use of optimal transport (OT) between a large noisy training dataset and a clean validation set, to value training data efficiently, without the dependency on model performance. However, the *LAVA* algorithm requires the entire dataset as an input, this limits its application to larger datasets. Inspired by the scalability of stochastic (gradient) approaches which carry out computations on *batches* of data points instead of the entire dataset, we analogously propose *SAVA*, a scalable variant of *LAVA* with its computation on batches of data points. Intuitively, *SAVA* follows the same scheme as *LAVA* which leverages the hierarchically defined OT for data valuation. However, while *LAVA* processes the whole dataset, *SAVA* divides the dataset into batches of data points, and carries out the OT problem computation on those batches. Moreover, our theoretical derivations on the trade-off of using entropic regularization for OT problems include refinements of prior work. We perform extensive experiments, to demonstrate that *SAVA* can scale to large datasets with millions of data points and does not trade off data valuation performance. Our Github repository is available at \url{https://github.com/skezle/sava}.

Distributionally robust optimization has emerged as an attractive way to train robust machine learning models, capturing data uncertainty and distribution shifts. Recent statistical analyses have proved that generalization guarantees of robust models based on the Wasserstein distance have generalization guarantees that do not suffer from the curse of dimensionality. However, these results are either approximate, obtained in specific cases, or based on assumptions difficult to verify in practice. In contrast, we establish exact generalization guarantees that cover a wide range of cases, with arbitrary transport costs and parametric loss functions, including deep learning objectives with nonsmooth activations. We complete our analysis with an excess bound on the robust objective and an extension to Wasserstein robust models with entropic regularizations.

Sliced Optimal Transport (OT) simplifies the OT problem in high-dimensional spaces by projecting supports of input measures onto one-dimensional lines, then exploiting the closed-form expression of the univariate OT to reduce the computational burden of OT. Recently, the Tree-Sliced method has been introduced to replace these lines with more intricate structures, known as tree systems. This approach enhances the ability to capture topological information of integration domains in Sliced OT while maintaining low computational cost. Inspired by this approach, in this paper, we present an adaptation of tree systems on OT problem for measures supported on a sphere. As counterpart to the Radon transform variant on tree systems, we propose a novel spherical Radon transform, with a new integration domain called spherical trees. By leveraging this transform and exploiting the spherical tree structures, we derive closed-form expressions for OT problems on the sphere. Consequently, we obtain an efficient metric for measures on the sphere, named Spherical Tree-Sliced Wasserstein (STSW) distance. We provide an extensive theoretical analysis to demonstrate the topology of spherical trees, the well-definedness and injectivity of our Radon transform variant, which leads to an orthogonally invariant distance between spherical measures. Finally, we conduct a wide range of numerical experiments, including gradient flows and self-supervised learning, to assess the performance of our proposed metric, comparing it to recent benchmarks.

To overcome computational challenges of Optimal Transport (OT), several variants of Sliced Wasserstein (SW) has been developed in the literature. These approaches exploit the closed-form expression of the univariate OT by projecting measures onto one-dimensional lines. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL) has emerged as a promising alternative that replaces these lines with a more intricate structure called tree systems. The tree structures enhance the ability to capture topological information of the metric while preserving computational efficiency. However, at the core of TSW-SL, the splitting maps, which serve as the mechanism for pushing forward measures onto tree systems, focus solely on the position of the measure supports while disregarding the projecting domains. Moreover, the specific splitting map used in TSW-SL leads to a metric that is not invariant under Euclidean transformations, a typically expected property for OT on Euclidean space. In this work, we propose a novel class of splitting maps that generalizes the existing one studied in TSW-SL enabling the use of all positional information from input measures, resulting in a novel Distance-based Tree-Sliced Wasserstein (*Db-TSW*) distance. In addition, we introduce a simple tree sampling process better suited for Db-TSW, leading to an efficient GPU-friendly implementation for tree systems, similar to the original SW. We also provide a comprehensive theoretical analysis of proposed class of splitting maps to verify the injectivity of the corresponding Radon Transform, and demonstrate that Db-TSW is an Euclidean invariant metric. We empirically show that Db-TSW significantly improves accuracy compared to recent SW variants while maintaining low computational cost via a wide range of experiments on gradient flows, image style transfer, and generative models.

We investigate the Sobolev IPM problem for probability measures supported on a graph metric space. Sobolev IPM is an important instance of integral probability metrics (IPM), and is obtained by constraining a critic function within a unit ball defined by the Sobolev norm. In particular, it has been used to compare probability measures and is crucial for several theoretical works in machine learning. However, to our knowledge, there are no efficient algorithmic approaches to compute Sobolev IPM effectively, which hinders its practical applications. In this work, we establish a relation between Sobolev norm and weighted $L^p$-norm, and leverage it to propose a novel regularization for Sobolev IPM. By exploiting the graph structure, we demonstrate that the regularized Sobolev IPM provides a closed-form expression for fast computation. This advancement addresses long-standing computational challenges, and paves the way to apply Sobolev IPM for practical applications, even in large-scale settings. Additionally, the regularized Sobolev IPM is negative definite. Utilizing this property, we design positive-definite kernels upon the regularized Sobolev IPM, and provide preliminary evidences of their advantages for comparing probability measures on a given graph for document classification and topological data analysis.

Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and leveraging the closed-form expression of the univariate OT to reduce the computational burden. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, in this work, we propose to replace one-dimensional lines with a more intricate structure, called tree systems. This structure is metrizable by a tree metric, which yields a closed-form expression for OT problems on tree systems. We provide an extensive theoretical analysis to formally define tree systems with their topological properties, introduce the concept of splitting maps, which operate as the projection mechanism onto these structures, then finally propose a novel variant of Radon transform for tree systems and verify its injectivity. This framework leads to an efficient metric between measures, termed Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL). By conducting a variety of experiments on gradient flows, image style transfer, and generative models, we illustrate that our proposed approach performs favorably compared to SW and its variants.

Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures. This approach enhances the ability to capture the topological structures of integration domains in Sliced Optimal Transport while maintaining low computational costs. Building on this foundation, we propose a novel nonlinear projectional framework for the Tree-Sliced Wasserstein (TSW) distance, substituting the linear projections in earlier versions with general projections, while ensuring the injectivity of the associated Radon Transform and preserving the well-definedness of the resulting metric. By designing appropriate projections, we construct efficient metrics for measures on both Euclidean spaces and spheres. Finally, we validate our proposed metric through extensive numerical experiments for Euclidean and spherical datasets. Applications include gradient flows, self-supervised learning, and generative models, where our methods demonstrate significant improvements over recent SW and TSW variants.

We investigate optimal transport (OT) for measures on graph metric spaces with different total masses. To mitigate the limitations of traditional $L^p$ geometry, Orlicz-Wasserstein (OW) and generalized Sobolev transport (GST) employ \emph{Orlicz geometric structure}, leveraging convex functions to capture nuanced geometric relationships and remarkably contribute to advance certain machine learning approaches. However, both OW and GST are restricted to measures with equal total mass, limiting their applicability to real-world scenarios where mass variation is common, and input measures may have noisy supports, or outliers. To address unbalanced measures, OW can either incorporate mass constraints or marginal discrepancy penalization, but this leads to a more complex two-level optimization problem. Additionally, GST provides a scalable yet rigid framework, which poses significant challenges to extend GST to accommodate nonnegative measures. To tackle these challenges, in this work we revisit the entropy partial transport (EPT) problem. By exploiting Caffarelli \& McCann's insights, we develop a novel variant of EPT endowed with Orlicz geometric structure, called \emph{Orlicz-EPT}. We establish theoretical background to solve Orlicz-EPT using a binary search algorithmic approach. Especially, by leveraging the dual EPT and the underlying graph structure, we formulate a novel regularization approach that leads to the proposed \emph{Orlicz-Sobolev transport} (OST). Notably, we demonstrate that OST can be efficiently computed by simply solving a univariate optimization problem, in stark contrast to the intensive computation needed for Orlicz-EPT. Building on this, we derive geometric structures for OST and draw its connections to other transport distances. We empirically illustrate that OST is several-order faster than Orlicz-EPT. Furthermore, we show preliminary evidence on the advantages of OST for measures on a graph in document classification and topological data analysis.

Optimal Transport (OT) has emerged as a fundamental tool in machine learning for comparing probability distributions in a geometrically meaningful manner. However, a key limitation of classical OT is its requirement that the source and target distributions have equal total mass, limiting its use in real-world settings involving imbalanced data, noise, outliers, or structural inconsistencies. Partial Transport (PT) addresses this limitation by allowing only a fraction of the mass to be transported, offering greater flexibility and robustness. Nonetheless, similar to OT, PT remains computationally expensive, as it typically involves solving large-scale linear programs—especially in high-dimensional spaces. To alleviate this computational burden, several emerging works have introduced the Tree-Sliced Wasserstein (TSW) distance, which projects distributions onto tree-metric spaces where OT problems admit closed-form solutions. Building on this line of research, we propose a novel framework that extends the tree-sliced approach to the PT setting, introducing the Partial Tree-Sliced Wasserstein (PartialTSW) distance. Our method is based on the key observation that, within tree-metric space, the PT problem can be equivalently reformulated as a standard balanced OT problem between suitably modified measures. This reformulation enables efficient computation while preserving the adaptability and robustness of partial transport. Our method proves effective across challenging tasks such as outlier removal and addressing class imbalance in image-to-image translation. Our code is publicly available at [this https URL](https://github.com/thanhqt2002/PartialTSW).

We study optimal transport (OT) problem for probability measures supported on a tree metric space. It is known that such OT problem (i.e., tree-Wasserstein (TW)) admits a closed-form expression, but depends fundamentally on the underlying tree structure over supports of input measures. In practice, the given tree structure may be, however, perturbed due to noisy or adversarial measurements. To mitigate this issue, we follow the max-min robust OT approach which considers the maximal possible distances between two input measures over an uncertainty set of tree metrics. In general, this approach is hard to compute, even for measures supported in one-dimensional space, due to its non-convexity and non-smoothness which hinders its practical applications, especially for large-scale settings. In this work, we propose novel uncertainty sets of tree metrics from the lens of edge deletion/addition which covers a diversity of tree structures in an elegant framework. Consequently, by building upon the proposed uncertainty sets, and leveraging the tree structure over supports, we show that the robust OT also admits a closed-form expression for a fast computation as its counterpart standard OT (i.e., TW). Furthermore, we demonstrate that the robust OT satisfies the metric property and is negative definite. We then exploit its negative definiteness to propose positive definite kernels and test them in several simulations on various real-world datasets on document classification and topological data analysis.

We study the optimal transport (OT) problem for measures supported on a graph metric space. Recently, Le et al. (2022) leverage the graph structure and propose a variant of OT, namely Sobolev transport (ST), which yields a closed-form expression for a fast computation. However, ST is essentially coupled with the $L^p$ geometric structure within its definition which makes it nontrivial to utilize ST for other prior structures. In contrast, the classic OT has the flexibility to adapt to various geometric structures by modifying the underlying cost function. An important instance is the Orlicz-Wasserstein (OW) which moves beyond the $L^p$ structure by leveraging the Orlicz geometric structure. Comparing to the usage of standard $p$-order Wasserstein, OW remarkably helps to advance certain machine learning approaches. Nevertheless, OW brings up a new challenge on its computation due to its two-level optimization formulation. In this work, we leverage a specific class of convex functions for Orlicz structure to propose the generalized Sobolev transport (GST). GST encompasses the ST as its special case, and can be utilized for prior structures beyond the $L^p$ geometry. In connection with the OW, we show that one only needs to simply solve a univariate optimization problem to compute the GST, unlike the complex two-level optimization problem in OW. We empirically illustrate that GST is several-order faster than the OW. Moreover, we provide preliminary evidences on the advantages of GST for document classification and for several tasks in topological data analysis.

Slicing distribution selection has been used as an effective technique to improve the performance of parameter estimators based on minimizing sliced Wasserstein distance in applications. Previous works either utilize expensive optimization to select the slicing distribution or use slicing distributions that require expensive sampling methods. In this work, we propose an optimization-free slicing distribution that provides a fast sampling for the Monte Carlo estimation of expectation. In particular, we introduce the random-path projecting direction (RPD) which is constructed by leveraging the normalized difference between two random vectors following the two input measures. From the RPD, we derive the random-path slicing distribution (RPSD) and two variants of sliced Wasserstein, i.e., the Random-Path Projection Sliced Wasserstein (RPSW) and the Importance Weighted Random-Path Projection Sliced Wasserstein (IWRPSW). We then discuss the topological, statistical, and computational properties of RPSW and IWRPSW. Finally, we showcase the favorable performance of RPSW and IWRPSW in gradient flow and the training of denoising diffusion generative models on images.

Optimal transport (OT) is a popular and powerful tool for comparing probability measures. However, OT suffers a few drawbacks: (i) input measures required to have the same mass, (ii) a high computational complexity, and (iii) indefiniteness which limits its applications on kernel-dependent algorithmic approaches. To tackle issues (ii)–(iii), Le et al. (2022) recently proposed Sobolev transport for measures on a graph having the same total mass by leveraging the graph structure over supports. In this work, we consider measures that may have different total mass and are supported on a graph metric space. To alleviate the disadvantages (i)–(iii) of OT, we propose a novel and scalable approach to extend Sobolev transport for this unbalanced setting where measures may have different total mass. We show that the proposed unbalanced Sobolev transport (UST) admits a closed-form formula for fast computation, and it is also negative definite. Additionally, we derive geometric structures for the UST and establish relations between our UST and other transport distances. We further exploit the negative definiteness to design positive definite kernels and evaluate them on various simulations to illustrate their fast computation and comparable performances against other transport baselines for unbalanced measures on a graph.

The scarcity of labeled data is a long-standing challenge for many machine learning tasks. We propose our gradient flow method to leverage the existing dataset (i.e., source) to generate new samples that are close to the dataset of interest (i.e., target). We lift both datasets to the space of probability distributions on the feature-Gaussian manifold, and then develop a gradient flow method that minimizes the maximum mean discrepancy loss. To perform the gradient flow of distributions on the curved feature-Gaussian space, we unravel the Riemannian structure of the space and compute explicitly the Riemannian gradient of the loss function induced by the optimal transport metric. For practical applications, we also propose a discretized flow, and provide conditional results guaranteeing the global convergence of the flow to the optimum. We illustrate the results of our proposed gradient flow method on several real-world datasets and show our method can improve the accuracy of classification models in transfer learning settings.

Optimal transport (OT) is a popular measure to compare probability distributions. However, OT suffers a few drawbacks such as (i) a high complexity for computation, (ii) indefiniteness which limits its applicability to kernel machines. In this work, we consider probability measures supported on a graph metric space and propose a novel Sobolev transport metric. We show that the Sobolev transport metric yields a closed-form formula for fast computation and it is negative definite. We show that the space of probability measures endowed with this transport distance is isometric to a bounded convex set in a Euclidean space with a weighted l_p distance. We further exploit the negative definiteness of the Sobolev transport to design positive-definite kernels, and evaluate their performances against other baselines in document classification with word embeddings and in topological data analysis.

Gromov-Wasserstein (GW) is a powerful tool to compare probability measures whose supports are in different metric spaces. However, GW suffers from a computational drawback since it requires to solve a complex non-convex quadratic program. In this work, we consider a specific family of cost metrics, namely, tree metrics for supports of each probability measure, to develop efficient and scalable discrepancies between the probability measures. Leveraging a tree structure, we propose to align flows from a root to each support instead of pair-wise tree metrics of supports, i.e., flows from a support to another support, in GW. Consequently, we propose a novel discrepancy, named Flow-based Alignment (FlowAlign), by matching the flows of the probability measures. FlowAlign is computationally fast and scalable for large-scale applications. Further exploring the tree structure, we propose a variant of FlowAlign, named Depth-based Alignment (DepthAlign), by aligning the flows hierarchically along each depth level of the tree structures. Theoretically, we prove that both FlowAlign and DepthAlign are pseudo-metrics. We also derive tree-sliced variants of the proposed discrepancies for applications without prior knowledge about tree structures for probability measures, computed by averaging FlowAlign/DepthAlign using random tree metrics, adaptively sampled from supports of probability measures. Empirically, we test our proposed approaches against other variants of GW baselines on a few benchmark tasks.

Optimal transport (OT) theory provides powerful tools to compare probability measures. However, OT is limited to nonnegative measures having the same mass, and suffers serious drawbacks about its computation and statistics. This leads to several proposals of regularized variants of OT in the recent literature. In this work, we consider an entropy partial transport (EPT) problem for nonnegative measures on a tree having different masses. The EPT is shown to be equivalent to a standard complete OT problem on a one-node extended tree. We derive its dual formulation, then leverage this to propose a novel regularization for EPT which admits fast computation and negative definiteness. To our knowledge, the proposed regularized EPT is the first approach that yields a closed-form solution among available variants of unbalanced OT for general nonnegative measures. For practical applications without prior knowledge about the tree structure for measures, we propose tree-sliced variants of the regularized EPT, computed by averaging the regularized EPT between these measures using random tree metrics, built adaptively from support data points. Exploiting the negative definiteness of our regularized EPT, we introduce a positive definite kernel, and evaluate it against other baselines on benchmark tasks such as document classification with word embedding and topological data analysis. In addition, we empirically demonstrate that our regularization also provides effective approximations.

Learning an effective representation of 3D point clouds requires a good metric to measure the discrepancy between two 3D point sets, which is non-trivial due to their irregularity. Most of the previous works resort to using the Chamfer discrepancy or Earth Mover’s distance, but those metrics are either ineffective in measuring the differences between point clouds or computationally expensive. In this paper, we conduct a systematic study with extensive experiments on distance metrics for 3D point clouds. From this study, we propose to use sliced Wasserstein distance and its variants for learning representations of 3D point clouds. In addition, we introduce a new algorithm to estimate sliced Wasserstein distance that guarantees that the estimated value is close enough to the true one. Experiments show that the sliced Wasserstein distance and its variants allow the neural network to learn a more efficient representation compared to the Chamfer discrepancy. We demonstrate the efficiency of the sliced Wasserstein metric and its variants on several tasks in 3D computer vision including training a point cloud autoencoder, generative modeling, transfer learning, and point cloud registration.

Neural architecture search (NAS) automates the design of deep neural networks. One of the main challenges in searching complex and non-continuous architectures is to compare the similarity of networks that the conventional Euclidean metric may fail to capture. Optimal transport (OT) is resilient to such complex structure by considering the minimal cost for transporting a network into another. However, the OT is generally not negative definite which may limit its ability to build the positive-definite kernels required in many kernel-dependent frameworks. Building upon tree-Wasserstein (TW), which is a negative definite variant of OT, we develop a novel discrepancy for neural architectures, and demonstrate it within a Gaussian process surrogate model for the sequential NAS settings. Furthermore, we derive a novel parallel NAS, using quality k-determinantal point process on the GP posterior, to select diverse and high-performing architectures from a discrete set of candidates. Empirically, we demonstrate that our TW-based approaches outperform other baselines in both sequential and parallel NAS.

In view of training increasingly complex learning architectures, we establish a nonsmooth implicit function theorem with an operational calculus. Our result applies to most practical problems (i.e., definable problems) provided that a nonsmooth form of the classical invertibility condition is fulfilled. This approach allows for formal subdifferentiation: for instance, replacing derivatives by Clarke Jacobians in the usual differentiation formulas is fully justified for a wide class of nonsmooth problems. Moreover this calculus is entirely compatible with algorithmic differentiation (e.g., backpropagation). We provide several applications such as training deep equilibrium networks, training neural nets with conic optimization layers, or hyperparameter-tuning for nonsmooth Lasso-type models. To show the sharpness of our assumptions, we present numerical experiments showcasing the extremely pathological gradient dynamics one can encounter when applying implicit algorithmic differentiation without any hypothesis.

Many machine learning tasks that involve predicting an output response can be solved by training a weighted regression model. Unfortunately, the predictive power of this type of models may severely deteriorate under low sample sizes or under covariate perturbations. Reweighting the training samples has aroused as an effective mitigation strategy to these problems. In this paper, we propose a novel and coherent scheme for kernel-reweighted regression by reparametrizing the sample weights using a doubly non-negative matrix. When the weighting matrix is confined in an uncertainty set using either the log-determinant divergence or the Bures-Wasserstein distance, we show that the adversarially reweighted estimate can be solved efficiently using first-order methods. Numerical experiments show that our reweighting strategy delivers promising results on numerous datasets.

Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of existing Bayesian optimization algorithms can only handle vector data and cannot handle complex structured data. In this paper, we propose the topological Bayesian optimization, which can efficiently find an optimal solution from structured data using \emph{topological information}. More specifically, in order to apply Bayesian optimization to structured data, we extract useful topological information from a structure and measure the proper similarity between structures. To this end, we utilize persistent homology, which is a topological data analysis method that was recently applied in machine learning. Moreover, we propose the Bayesian optimization algorithm that can handle multiple types of topological information by using a linear combination of kernels for persistence diagrams. Through experiments, we show that topological information extracted by persistent homology contributes to a more efficient search for optimal structures compared to the random search baseline and the graph Bayesian optimization algorithm.

Popular machine learning estimators involve regularization parameters that can be challenging to tune, and standard strategies rely on grid search for this task. In this paper, we revisit the techniques of approximating the regularization path up to predefined tolerance $\epsilon$ in a unified framework and show that its complexity is $O(1/\sqrt[d]{\epsilon})$ for uniformly convex loss of order $d \geq 2$ and $O(1/\sqrt{\epsilon})$ for Generalized Self-Concordant functions. This framework encompasses least-squares but also logistic regression, a case that as far as we know was not handled as precisely in previous works. We leverage our technique to provide refined bounds on the validation error as well as a practical algorithm for hyperparameter tuning. The latter has global convergence guarantee when targeting a prescribed accuracy on the validation set. Last but not least, our approach helps relieving the practitioner from the (often neglected) task of selecting a stopping criterion when optimizing over the training set: our method automatically calibrates this criterion based on the targeted accuracy on the validation set.

Automated tasks play an important role in both consumer and industrial IoT environments. In many scenarios, the IoT tasks are performed given certain conditions. To facilitate the tasks, it is necessary to delegate a third party to listen to events that trigger the conditions and issue commands to the IoT resources accordingly. However, without restriction, the third party may be over-privileged and able to control the resources unconditionally. We define the third party's permission to act under some conditions as a conditional action. We propose PPCA, a privacy-preserving service that allows users to create conditional actions in a decentralized platform using smart contracts. PPCA can guarantee strict privilege at the third party that holds conditional actions. By generalizing a variety of conditions into simple forms of conditional logic, the conditions can be verified in a privacy-preserving manner. We build a prototype of PPCA on Ethereum. The performance shows the feasibility of PPCA in IoT environments.

Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants of OT in the recent literature, one of the most notable being the \textit{sliced} formulation, which exploits the closed-form formula between univariate distributions by projecting high-dimensional measures onto random lines. We consider in this work a more general family of ground metrics, namely \textit{tree metrics}, which also yield fast closed-form computations and negative definite, and of which the sliced-Wasserstein distance is a particular case (the tree is a chain). We propose the tree-sliced Wasserstein distance, computed by averaging the Wasserstein distance between these measures using random tree metrics, built adaptively in either low or high-dimensional spaces. Exploiting the negative definiteness of that distance, we also propose a positive definite kernel, and test it against other baselines on a few benchmark tasks.

Secure access control to a wide variety of Internet of Things (IoT)devices has become critical. Blockchain-based access control frameworks are promising technologies to support secure access to IoT devices in pervasive computing applications. However, in most of the proposed solutions, the IoT devices rely on a trusted server to retrieve critical access control data from the blockchains. We propose a method for IoT devices to validate blockchain data without solely being dependent on a central server. In our approach, several witnesses on the network can be selected randomly by the devices to validate access control information. Our method is aided by Bloom filters, which are shown to be lightweight for resource-constrained devices.

Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data such as shapes, linked twist maps, and material data. Among them, \textit{persistent homology} is a well-known tool to extract robust topological features, and outputs as \textit{persistence diagrams} (PDs). However, PDs are point multi-sets which can not be used in machine learning algorithms for vector data. To deal with it, an emerged approach is to use kernel methods, and an appropriate geometry for PDs is an important factor to measure the similarity of PDs. A popular geometry for PDs is the \textit{Wasserstein metric}. However, Wasserstein distance is not \textit{negative definite}. Thus, it is limited to build positive definite kernels upon the Wasserstein distance \textit{without approximation}. In this work, we rely upon the alternative \textit{Fisher information geometry} to propose a positive definite kernel for PDs \textit{without approximation}, namely the Persistence Fisher (PF) kernel. Then, we analyze eigensystem of the integral operator induced by the proposed kernel for kernel machines. Based on that, we derive generalization error bounds via covering numbers and Rademacher averages for kernel machines with the PF kernel. Additionally, we show some nice properties such as stability and infinite divisibility for the proposed kernel. Furthermore, we also propose a linear time complexity over the number of points in PDs for an approximation of our proposed kernel with a bounded error. Throughout experiments with many different tasks on various benchmark datasets, we illustrate that the PF kernel compares favorably with other baseline kernels for PDs.

Devices to support pervasive computing and the Internet of Things (IoT) are becoming present in almost every aspect of our lives. Due to their limited power and computation, these devices often need to rely on some more powerful outsourced cloud services, which raises a security and privacy concern as IoT data is often sensitive. On the other hand, blockchain technology has recently gained much attention due to its decentralized, trustless and immutable design. We propose CapChain - an access control framework based on blockchain that allows users to share and delegate their access rights easily to IoT devices in public but still maintain privacy. To protect privacy, we adapt multiple techniques from anonymous crypto-currency blockchain systems to hide sensitive information, including users' identities and related information about the capabilities. We also build a testbed as a proof of concept.

Many applications in machine learning handle bags of features or histograms rather than simple vectors. In that context, defining a proper geometry to compare histograms can be crucial for many machine learning algorithms. While one might be tempted to use a default metric such as the Euclidean metric, empirical evidence shows this may not be the best choice when dealing with observations that lie in the probability simplex. Additionally, it might be desirable to choose a metric adaptively based on data. We consider in this paper the problem of learning a Riemannian metric on the simplex given unlabeled histogram data. We follow the approach of Lebanon(2006), who proposed to estimate such a metric within a parametric family by maximizing the inverse volume of a given data set of points under that metric. The metrics we consider on the multinomial simplex are pull-back metrics of the Fisher information parameterized by operations within the simplex known as Aitchison(1982) transformations. We propose an algorithmic approach to maximize inverse volumes using sampling and contrastive divergences. We provide experimental evidence that the metric obtained under our proposal outperforms alternative approaches.

Learning distances that are specifically designed to compare histograms in the probability simplex has recently attracted the attention of the community. Learning such distances is important because most machine learning problems involve bags of features rather than simple vectors. Ample empirical evidence suggests that the Euclidean distance in general and Mahalanobis metric learning in particular may not be suitable to quantify distances between points in the simplex. We propose in this paper a new contribution to address this problem by generalizing a family of embeddings proposed by Aitchison (1982) to map the probability simplex onto a suitable Euclidean space. We provide algorithms to estimate the parameters of such maps, and show that these algorithms lead to representations that outperform alternative approaches to compare histograms.