In this paper, we propose a flow-based method for learning all-to-all transfer maps among conditional distributions that approximates pairwise optimal transport. The proposed method addresses the challenge of handling the case of continuous conditions, which often involve a large set of conditions with sparse empirical observations per condition. We introduce a novel cost function that enables simultaneous learning of optimal transports for all pairs of conditional distributions. Our method is supported by a theoretical guarantee that, in the limit, it converges to the pairwise optimal transports among infinite pairs of conditional distributions. The learned transport maps are subsequently used to couple data points in conditional flow matching. We demonstrate the effectiveness of this method on synthetic and benchmark datasets, as well as on chemical datasets in which continuous physical properties are defined as conditions.

Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We present the theoretical foundations of NFT and show that the existence of a linear equivariant feature, which has been assumed ubiquitously in equivariance learning, is equivalent to the existence of a group invariant kernel on the dataspace. We also provide experimental results to demonstrate the application of NFT in typical scenarios with varying levels of knowledge about the acting group.

In this paper, we introduce a novel method for generating fractional time series through the utilization of neural networks. Although Neural Stochastic Differential Equations (Neural SDEs) have been presented as a method that combines Deep Neural Networks with numerical solvers of differential equations, these typically presume the noise structure of standard Brownian motion (Bm). Contrarily, numerous real-world time series data exhibit a fractal property, characterized by a Hurst index (H) that ranges from 0 to 1. This type of fractional time series pervades various domains including physics, biology, hydrology, network research, and financial mathematics. We propose a Latent Fractional Net (Lf-Net), devised to encapsulate both the long-range dependence (H > 1/2) and roughness (H < 1/2) intrinsic to fractional time series. This is accomplished by augmenting the noise term of the Neural SDEs using fractional Brownian motion (fBm) with an arbitrary Hurst index. We prove the existence and uniqueness of the solutions of the Lf-Net and theoretically show the convergence of the numerical solutions. We demonstrate the robustness of the Lf-Net under proper nonlinear transformations and construct a generative model for time-series data. The experiments show that the calibrated generator of the model can replicate the distributional properties of the original time series, especially the Hurst index. We conclude that our Lf-Net can effectively model the complex noise structure of real-world time series data and provide a promising direction for time series data generation.

Lookup tables (LUTs) are commonly used to speed up image processing by handling complex mathematical functions like sine and exponential calculations. They are used in various applications such as camera image processing, high-dynamic range imaging, and edge-preserving filtering. However, due to the increasing gap between computing and input/output performance, LUTs are becoming less effective. Even though specific circuits like SIMD can improve LUT efficiency, they still need to bridge the performance gap fully. The gap makes it difficult to choose between direct numerical and LUT calculations. For this problem, a register-LUTs method with the nearest neighbor was proposed; however, it is limited for functions with narrow-range values approaching zero. In this paper, we propose a method for using register LUTs to process images efficiently over a wide range of values. Our contributions include proposing register-LUT with linear interpolation for efficient computation, using a smaller data type for further efficiency, and suggesting an efficient data retrieving method.

In this paper, we focus on the generation of time-series data using neural networks. It is often the case that input time-series data have only one realized (and usually irregularly sampled) path, which makes it difficult to extract time-series characteristics, and its noise structure is more complicated than i.i.d. type. Time series data, especially from hydrology, telecommunications, economics, and finance, exhibit long-term memory also called long-range dependency (LRD). The main purpose of this paper is to artificially generate time series with the help of neural networks, making the LRD of paths into account. We propose fSDE-Net: neural fractional Stochastic Differential Equation Network. It generalizes the neural stochastic differential equation model by using fractional Brownian motion with a Hurst index larger than half, which exhibits the LRD property. We derive the solver of fSDE-Net and theoretically analyze the existence and uniqueness of the solution to fSDE-Net. Our experiments with artificial and real time-series data demonstrate that the fSDE-Net model can replicate distributional properties well.

In this paper, we study random subsampling of Gaussian process regression, one of the simplest approximation baselines, from a theoretical perspective. Although subsampling discards a large part of training data, we show provable guarantees on the accuracy of the predictive mean/variance and its generalization ability.For analysis, we consider embedding kernel matrices into graphons, which encapsulate the difference of the sample size and enables us to evaluate the approximation and generalization errors in a unified manner. The experimental results show that the subsampling approximation achieves a better trade-off regarding accuracy and runtime than the ystrom and random Fourier expansion methods.

Tensor decomposition methods are widely used for model compression and fast inference in convolutional neural networks (CNNs). Although many decompositions are conceivable, only CP decomposition and a few others have been applied in practice, and no extensive comparisons have been made between available methods. Previous studies have not determined how many decompositions are available, nor which of them is optimal. In this study, we first characterize a decomposition class specific to CNNs by adopting a flexible graphical notation. The class includes such well-known CNN modules as depthwise separable convolution layers and bottleneck layers, but also previously unknown modules with nonlinear activations. We also experimentally compare the tradeoff between prediction accuracy and time/space complexity for modules found by enumerating all possible decompositions, or by using a neural architecture search. We find some nonlinear decompositions outperform existing ones.

Tree ensembles, such as random forests, are renowned for their high prediction performance. However, their interpretability is critically limited due to the enormous complexity. In this study, we propose a method to make a complex tree ensemble interpretable by simplifying the model. Specifically, we formalize the simplification of tree ensembles as a model selection problem. Given a complex tree ensemble, we aim at obtaining the simplest representation that is essentially equivalent to the original one. To this end, we derive a Bayesian model selection algorithm that optimizes the simplified model while maintaining the prediction performance. Our numerical experiments on several datasets showed that complicated tree ensembles were approximated interpretably.

Representing the semantic relations that exist between two given words (or entities) is an important first step in a wide-range of NLP applications such as analogical reasoning, knowledge base completion and relational information retrieval. A simple, yet surprisingly accurate method for representing a relation between two words is to compute the vector offset (PairDiff) between their corresponding word embeddings. Despite the empirical success, it remains unclear as to whether PairDiff is the best operator for obtaining a relational representation from word embeddings. We conduct a theoretical analysis of generalised bilinear operators that can be used to measure the l2 relational distance between two word-pairs. We show that, if the word embed- dings are standardised and uncorrelated, such an operator will be independent of bilinear terms, and can be simplified to a linear form, where PairDiff is a special case. For numerous word embedding types, we empirically verify the uncorrelation assumption, demonstrating the general applicability of our theoretical result. Moreover, we experimentally discover PairDiff from the bilinear relational compositional operator on several benchmark analogy datasets.

Distributed word embeddings have shown superior performances in numerous Natural Language Processing (NLP) tasks. However, their performances vary significantly across different tasks, implying that the word embeddings learnt by those methods capture complementary aspects of lexical semantics. Therefore, we believe that it is important to combine the existing word embeddings to produce more accurate and complete meta-embeddings of words. For this purpose, we propose an unsupervised locally linear meta-embedding learning method that takes pre-trained word embeddings as the input, and produces more accurate meta embeddings. Unlike previously proposed meta-embedding learning methods that learn a global projection over all words in a vocabulary, our proposed method is sensitive to the differences in local neighbourhoods of the individual source word embeddings. Moreover, we show that vector concatenation, a previously proposed highly competitive baseline approach for integrating word embeddings, can be derived as a special case of the proposed method. Experimental results on semantic similarity, word analogy, relation classification, and short-text classification tasks show that our meta-embeddings to significantly outperform prior methods in several benchmark datasets, establishing a new state of the art for meta-embeddings.

Real data tensors are usually high dimensional but their intrinsic information is preserved in low-dimensional space, which motivates to use tensor decompositions such as Tucker decomposition. Often, real data tensors are not only low dimensional, but also smooth, meaning that the adjacent elements are similar or continuously changing, which typically appear as spatial or temporal data. To incorporate the smoothness property, we propose the smoothed Tucker decomposition (STD). STD leverages the smoothness by the sum of a few basis functions, which reduces the number of parameters. The objective function is formulated as a convex problem and, to solve that, an algorithm based on the alternating direction method of multipliers is derived. We theoretically show that, under the smoothness assumption, STD achieves a better error bound. The theoretical result and performances of STD are numerically verified.

How can we decompose a data tensor if the indices are partially missing?Tensor decomposition is a fundamental tool to analyze the tensor data.Suppose, for example, we have a 3rd-order tensor X where each element Xijk takes 1 if user i posts word j at location k on Twitter.Standard tensor decomposition expects all the indices are observed but, in some tweets, location k can be missing.In this paper, we study a tensor decomposition problem where the indices (i, j, or k) of some observed elements are partially missing.Towards the problem, we propose a probabilistic tensor decomposition model that handles missing indices as latent variables.To infer them, we derive an algorithm based on stochastic variational inference, which enables to leverage the information from the incomplete data scalably. The experiments on both synthetic and real datasets show that the proposed method achieves higher accuracy in the tensor completion task than baselines that cannot handle missing indices.

What kinds of data does Label Propagation (LP) work best on? Can we justify the solution of LP from a theoretical standpoint? LP is a semi-supervised learning algorithm that is widely used to predict unobserved node labels on a network (e.g., user's gender on an SNS). Despite its importance, its theoretical properties remain mostly unexplored. In this paper, we answer the above questions by interpreting LP from a statistical viewpoint. As our main result, we identify the network generative model behind the discretized version of LP (DLP), and we show that under specific conditions the solution of DLP is equal to the maximum {\it a posteriori} estimate of that generative model. Our main result reveals the critical limitations of LP. Specifically, we discover that LP would not work best on networks with (1) disassortative node labels, (2) clusters having different edge densities, (3) non-uniform label distributions, or (4) unreliable node labels provided. Our experiments under a variety of settings support our theoretical results.

In this paper, we develop an algorithm that approximates the residual error of Tucker decomposition, one of the most popular tensor decomposition methods, with a provable guarantee. Given an order-$K$ tensor $X\in\mathbb{R}^{N_1\times\cdots\times N_K}$, our algorithm randomly samples a constant number $s$ of indices for each mode and creates a ``mini'' tensor $\tilde{X}\in\mathbb{R}^{s\times\cdots\times s}$, whose elements are given by the intersection of the sampled indices on $X$. Then, we show that the residual error of the Tucker decomposition of $\tilde{X}$ is sufficiently close to that of $X$ with high probability. This result implies that we can figure out how much we can fit a low-rank tensor to $X$ \emph{in constant time}, regardless of the size of $X$. This is useful for guessing the favorable rank of Tucker decomposition. Finally, we demonstrate how the sampling method works quickly and accurately using multiple real datasets.

Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations. First, we introduce a convex relaxation of the TT decomposition problem and derive its error bound for the tensor completion task. Next, we develop a randomized optimization method, in which the time complexity is as efficient as the space complexity is. In experiments, we numerically confirm the derived bounds and empirically demonstrate the performance of our method with a real higher-order tensor.

In this study, we investigate expected CP decomposition — a special case of CP decomposition in which a tensor to be decomposed is given as the sum or average of tensor samples X(t) for t = 1,...,T. To determine this decomposition, we develope stochastic-gradient-descent-type algorithms with four appealing features: efficient memory use, ability to work in an online setting, robustness of parameter tuning, and simplicity. Our theoretical analysis show that the solutions do not diverge to infinity for any initial value or step size. Experimental results confirm that our algorithms significantly outperform all existing methods in terms of accuracy. We also show that they can successfully decompose a large tensor, containing billion-scale nonzero elements.

Nonparametric extension of tensor regression is proposed. Nonlinearity in a high-dimensional tensor space is broken into simple local functions by incorporating low-rank tensor decomposition. Compared to naive nonparametric approaches, our formulation considerably improves the convergence rate of estimation while maintaining consistency with the same function class under specific conditions. To estimate local functions, we develop a Bayesian estimator with the Gaussian process prior. Experimental results show its theoretical properties and high performance in terms of predicting a summary statistic of a real complex network.

A sampling-based optimization method for quadratic functions is proposed. Our method approximately solves the following $n$-dimensional quadratic minimization problem in constant time, which is independent of $n$: $z^*=\min_{\bv \in \bbR^n}\bracket{\bv}{A \bv} + n\bracket{\bv}{\diag(\bd)\bv} + n\bracket{\bb}{\bv}$, where $A \in \bbR^{n \times n}$ is a matrix and $\bd,\bb \in \bbR^n$ are vectors. Our theoretical analysis specifies the number of samples $k(\delta, \epsilon)$ such that the approximated solution $z$ satisfies $|z - z^*| = O(\epsilon n^2)$ with probability $1-\delta$. The empirical performance (accuracy and runtime) is positively confirmed by numerical experiments.

A fundamental yet new challenge in information retrieval is the identification of patterns behind search queries. For example, the query "NY restaurant" and "boston hotel" shares the common pattern "LOCATION SERVICE". However, because of the diversity of real queries, existing approaches require data preprocessing by humans or specifying the target query domains, which hinders their applicability.

Factorized information criterion (FIC) is a recently developed approximation technique for the marginal log-likelihood, which provides an automatic model selection framework for a few latent variable models (LVMs) with tractable inference algorithms. This paper reconsiders FIC and fills theoretical gaps of previous FIC studies. First, we reveal the core idea of FIC that allows generalization for a broader class of LVMs, including continuous LVMs, in contrast to previous FICs, which are applicable only to binary LVMs. Second, we investigate the model selection mechanism of the generalized FIC. Our analysis provides a formal justification of FIC as a model selection criterion for LVMs and also a systematic procedure for pruning redundant latent variables that have been removed heuristically in previous studies. Third, we provide an interpretation of FIC as a variational free energy and uncover previously-unknown their relationship. A demonstrative study on Bayesian principal component analysis is provided and numerical experiments support our theoretical results.

Twitter is a "what's-happening-right-now" tool that enables interested parties to follow thoughts and commentary of individual users in nearly real-time. While it is a valuable source of information for real-time topic detection and tracking, Twitter data are not clean because of noisy messages and users, which significantly diminish the reliability of obtained results.

This paper extends factorized asymptotic Bayesian (FAB) inference for latent feature models~(LFMs). FAB inference has not been applicable to models, including LFMs, without a specific condition on the Hesqsian matrix of a complete log-likelihood, which is required to derive a factorized information criterion''~(FIC). Our asymptotic analysis of the Hessian matrix of LFMs shows that FIC of LFMs has the same form as those of mixture models. FAB/LFMs have several desirable properties (e.g., automatic hidden states selection and parameter identifiability) and empirically perform better than state-of-the-art Indian Buffet processes in terms of model selection, prediction, and computational efficiency."

Reinforcement learning (RL) methods based on direct policy search (DPS) have been actively discussed to achieve an efficient approach to complicated Markov decision processes (MDPs). Although they have brought much progress in practical applications of RL, there still remains an unsolved problem in DPS related to model selection for the policy. In this paper, we propose a novel DPS method, {\it weighted likelihood policy search (WLPS)}, where a policy is efficiently learned through the weighted likelihood estimation. WLPS naturally connects DPS to the statistical inference problem and thus various sophisticated techniques in statistics can be applied to DPS problems directly. Hence, by following the idea of the {\it information criterion}, we develop a new measurement for model comparison in DPS based on the weighted log-likelihood.

Most of the existing analysis methods for tensors (or multi-way arrays) only assume that tensors to be completed are of low rank. However, for example, when they are applied to tensor completion problems, their prediction accuracy tends to be significantly worse when only a limited number of entries are observed. In this paper, we propose to use relationships among data as auxiliary information in addition to the low-rank assumption to improve the quality of tensor decomposition. We introduce two regularization approaches using graph Laplacians induced from the relationships, one for moderately sparse cases and the other for extremely sparse cases. We also give present two kinds of iterative algorithms for approximate solutions: one based on an EM-like algorithms which is stable but not so scalable, and the other based on gradient-based optimization which is applicable to large scale datasets. Numerical experiments on tensor completion using synthetic and benchmark datasets show that the use of auxiliary information improves completion accuracy over the existing methods based only on the low-rank assumption, especially when observations are sparse.

The increasing interest in dynamically changing networks has led to growing interest in a more general link prediction problem called temporal link prediction in the data mining and machine learning communities. However, only links in identical time frames are considered in temporal link prediction. We propose a new link prediction problem called cross-temporal link prediction in which the links among nodes in different time frames are inferred. A typical example of cross-temporal link prediction is cross-temporal entity resolution to determine the identity of real entities represented by data objects observed in different time periods. In dynamic environments, the features of data change over time, making it difficult to identify cross-temporal links by directly comparing observed data. Other examples of cross-temporal links are asynchronous communications in social networks such as Face book and Twitter, where a message is posted in reply to a previous message. We adopt a dimension reduction approach to cross-temporal link prediction, that is, data objects in different time frames are mapped into a common low-dimensional latent feature space, and the links are identified on the basis of the distance between the data objects. The proposed method uses different low-dimensional feature projections in different time frames, enabling it to adapt to changes in the latent features over time. Using multi-task learning, it jointly learns a set of feature projection matrices from the training data, given the assumption of temporal smoothness of the projections. The optimal solutions are obtained by solving a single generalized eigenvalue problem. Experiments using a real-world set of bibliographic data for cross-temporal entity resolution showed that introducing time-dependent feature projections improves the accuracy of link prediction.

We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as non-convex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared error of the convex method scales linearly with the quantity we call the normalized rank of the true tensor. The current analysis naturally extends the analysis of convex low-rank matrix estimation to tensors. Furthermore, we show through numerical experiments that our theory can precisely predict the scaling behaviour in practice.

In this paper, we study probabilistic modeling of heterogeneously attributed multi-dimensional arrays. The model can manage the heterogeneity by employing an individual exponential-family distribution for each attribute of the tensor array. These entries are connected by latent variables and are shared information across the different attributes. Because a Bayesian inference for our model is intractable, we cast the EM algorithm approximated by using the Lap lace method and Gaussian process. This approximation enables us to derive a predictive distribution for missing values in a consistent manner. Simulation experiments show that our method outperforms other methods such as PARAFAC and Tucker decomposition in missing-values prediction for cross-national statistics and is also applicable to discover anomalies in heterogeneous office-logging data.

Complex-valued Hopfield network is used to model the dynamics of a network of limit cycle oscillators, each of which emerges via Hopf bifurcation, to investigate the dependency of the network dynamics on a bifurcation parameter. As an application, a network of two complex-valued neurons is used as a central pattern generator model for a biped locomotion. A bifurcation parameter is a constant input from higher motor centers. Numerical calculations show the system successfully expresses some characteristic behaviors, which were obtained by more complicated Fitzhugh-Nagumo oscillator model, and which were found in clinical data of disordered interlimb coordination caused by Parkinson's disease. The observed results of symmetric anti -phase synchronization, asymmetric synchronization, and breakdown of the synchronization can be explained by the existence condition of the energy function of the complex-valued neural network, and by the synchronization condition of a coupled system of phase oscillators.