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Causal representation learning algorithms discover lower-dimensional representations of data that admit a decipherable interpretation of cause and effect; as achieving such interpretable representations is challenging, many causal learning algorithms utilize elements indicating prior information, such as (linear) structural causal models, interventional data, or weak supervision. Unfortunately, in exploratory causal representation learning, such elements and prior information may not be available or warranted. Alternatively, scientific datasets often have multiple modalities or physics-based constraints, and the use of such scientific, multimodal data has been shown to improve disentanglement in fully unsupervised settings. Consequently, we introduce a causal representation learning algorithm (causalPIMA) that can use multimodal data and known physics to discover important features with causal relationships. Our innovative algorithm utilizes a new differentiable parametrization to learn a directed acyclic graph (DAG) together with a latent space of a variational autoencoder in an end-to-end differentiable framework via a single, tractable evidence lower bound loss function. We place a Gaussian mixture prior on the latent space and identify each of the mixtures with an outcome of the DAG nodes; this novel identification enables feature discovery with causal relationships. Tested against a synthetic and a scientific dataset, our results demonstrate the capability of learning an interpretable causal structure while simultaneously discovering key features in a fully unsupervised setting.

We investigate the so-called "MMSE conjecture" from Guo et al. (2011) which asserts that two distributions on the real line with the same entropy along the heat flow coincide up to translation and symmetry. Our approach follows the path breaking contribution Ledoux (1995) which gave algebraic representations of the derivatives of said entropy in terms of multivariate polynomials. The main contributions in this note are (i) we obtain the leading terms in the polynomials from Ledoux (1995), and (ii) we provide new conditions on the source distributions ensuring the MMSE conjecture holds. As illustrating examples, our findings cover the cases of uniform and Rademacher distributions, for which previous results in the literature were inapplicable.

We give a categorical treatment, in the spirit of Baez and Fritz, of relative entropy for probability distributions defined on standard Borel spaces. We define a category suitable for reasoning about statistical inference on standard Borel spaces. We define relative entropy as a functor into Lawvere's category and we show convexity, lower semicontinuity and uniqueness.

This paper introduces a new mathematical and numerical framework for surface analysis derived from the general setting of elastic Riemannian metrics on shape spaces. Traditionally, those metrics are defined over the infinite dimensional manifold of immersed surfaces and satisfy specific invariance properties enabling the comparison of surfaces modulo shape preserving transformations such as reparametrizations. The specificity of the approach we develop is to restrict the space of allowable transformations to predefined finite dimensional bases of deformation fields. These are estimated in a data-driven way so as to emulate specific types of surface transformations observed in a training set. The use of such bases allows to simplify the representation of the corresponding shape space to a finite dimensional latent space. However, in sharp contrast with methods involving e.g. mesh autoencoders, the latent space is here equipped with a non-Euclidean Riemannian metric precisely inherited from the family of aforementioned elastic metrics. We demonstrate how this basis restricted model can be then effectively implemented to perform a variety of tasks on surface meshes which, importantly, does not assume these to be pre-registered (i.e. with given point correspondences) or to even have a consistent mesh structure. We specifically validate our approach on human body shape and pose data as well as human face scans, and show how it generally outperforms state-of-the-art methods on problems such as shape registration, interpolation, motion transfer or random pose generation.

We study versions of Hilbert's projective metric for spaces of integrable functions of bounded growth. These metrics originate from cones which are relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. We show that kernel integral operators are contractions with respect to suitable specifications of such metrics even for kernels which are not bounded away from zero, provided that the decay to zero of the kernel is controlled. As an application to entropic optimal transport, we show exponential convergence of Sinkhorn's algorithm in settings where the marginal distributions have sufficiently light tails compared to the growth of the cost function.

Imaging through perturbed multimode fibres based on deep learning has been widely researched. However, existing methods mainly use target-speckle pairs in different configurations. It is challenging to reconstruct targets without trained networks. In this paper, we propose a physics-assisted, unsupervised, learning-based fibre imaging scheme. The role of the physical prior is to simplify the mapping relationship between the speckle pattern and the target image, thereby reducing the computational complexity. The unsupervised network learns target features according to the optimized direction provided by the physical prior. Therefore, the reconstruction process of the online learning only requires a few speckle patterns and unpaired targets. The proposed scheme also increases the generalization ability of the learning-based method in perturbed multimode fibres. Our scheme has the potential to extend the application of multimode fibre imaging.

Completely random measures (CRMs) and their normalizations (NCRMs) offer flexible models in Bayesian nonparametrics. But their infinite dimensionality presents challenges for inference. Two popular finite approximations are truncated finite approximations (TFAs) and independent finite approximations (IFAs). While the former have been well-studied, IFAs lack similarly general bounds on approximation error, and there has been no systematic comparison between the two options. In the present work, we propose a general recipe to construct practical finite-dimensional approximations for homogeneous CRMs and NCRMs, in the presence or absence of power laws. We call our construction the automated independent finite approximation (AIFA). Relative to TFAs, we show that AIFAs facilitate more straightforward derivations and use of parallel computing in approximate inference. We upper bound the approximation error of AIFAs for a wide class of common CRMs and NCRMs -- and thereby develop guidelines for choosing the approximation level. Our lower bounds in key cases suggest that our upper bounds are tight. We prove that, for worst-case choices of observation likelihoods, TFAs are more efficient than AIFAs. Conversely, we find that in real-data experiments with standard likelihoods, AIFAs and TFAs perform similarly. Moreover, we demonstrate that AIFAs can be used for hyperparameter estimation even when other potential IFA options struggle or do not apply.

This is the second in a series of articles aimed at exploring the relationship between the complexity classes of P and NP. The research in this article aims to find conditions of an algorithmic nature that are necessary and sufficient to transform any Boolean function in conjunctive normal form into a specific form that guarantees the satisfiability of this function. To find such conditions, we use the concept of a special covering of a set introduced in [13], and investigate the connection between this concept and the notion of satisfiability of Boolean functions. As shown, the problem of existence of a special covering for a set is equivalent to the Boolean satisfiability problem. Thus, an important result is the proof of the existence of necessary and sufficient conditions that make it possible to find out if there is a special covering for the set under the special decomposition. This result allows us to formulate the necessary and sufficient algorithmic conditions for Boolean satisfiability, considering the function in conjunctive normal form as a set of clauses. In parallel, as a result of the aforementioned algorithmic procedure, we obtain the values of the variables that ensure the satisfiability of this function. The terminology used related to graph theory, set theory, Boolean functions and complexity theory is consistent with the terminology in [1], [2], [3], [4]. The newly introduced terms are not found in use by other authors and do not contradict to other terms.

This paper introduces a novel approach for multi-task regression that connects Kernel Machines (KMs) and Extreme Learning Machines (ELMs) through the exploitation of the Random Fourier Features (RFFs) approximation of the RBF kernel. In this sense, one of the contributions of this paper shows that for the proposed models, the KM and the ELM formulations can be regarded as two sides of the same coin. These proposed models, termed RFF-BLR, stand on a Bayesian framework that simultaneously addresses two main design goals. On the one hand, it fits multitask regressors based on KMs endowed with RBF kernels. On the other hand, it enables the introduction of a common-across-tasks prior that promotes multioutput sparsity in the ELM view. This Bayesian approach facilitates the simultaneous consideration of both the KM and ELM perspectives enabling (i) the optimisation of the RBF kernel parameter $\gamma$ within a probabilistic framework, (ii) the optimisation of the model complexity, and (iii) an efficient transfer of knowledge across tasks. The experimental results show that this framework can lead to significant performance improvements compared to the state-of-the-art methods in multitask nonlinear regression.

Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.