Kenji Yamanishi

Naoki Nishikawa, Yuichi Ike, Kenji Yamanishi

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.

Kenji Yamanishi, So Hirai

We address the issue of detecting changes of models that lie behind a data stream. The model refers to an integer-valued structural information such as the number of free parameters in a parametric model. Specifically we are concerned with the problem of how we can detect signs of model changes earlier than they are actualized. To this end, we employ {\em continuous model selection} on the basis of the notion of {\em descriptive dimensionality}~(Ddim). It is a real-valued model dimensionality, which is designed for quantifying the model dimensionality in the model transition period. Continuous model selection is to determine the real-valued model dimensionality in terms of Ddim from a given data. We propose a novel methodology for detecting signs of model changes by tracking the rise-up of Ddim in a data stream. We apply this methodology to detecting signs of changes of the number of clusters in a Gaussian mixture model and those of the order in an auto regression model. With synthetic and real data sets, we empirically demonstrate its effectiveness by showing that it is able to visualize well how rapidly model dimensionality moves in the transition period and to raise early warning signals of model changes earlier than they are detected with existing methods.

Shintaro Fukushima, Kenji Yamanishi

This study addresses the issue of balancing graph summarization and graph change detection. Graph summarization compresses large-scale graphs into a smaller scale. However, the question remains: To what extent should the original graph be compressed? This problem is solved from the perspective of graph change detection, aiming to detect statistically significant changes using a stream of summary graphs. If the compression rate is extremely high, important changes can be ignored, whereas if the compression rate is extremely low, false alarms may increase with more memory. This implies that there is a trade-off between compression rate in graph summarization and accuracy in change detection. We propose a novel quantitative methodology to balance this trade-off to simultaneously realize reliable graph summarization and change detection. We introduce a probabilistic structure of hierarchical latent variable model into a graph, thereby designing a parameterized summary graph on the basis of the minimum description length principle. The parameter specifying the summary graph is then optimized so that the accuracy of change detection is guaranteed to suppress Type I error probability (probability of raising false alarms) to be less than a given confidence level. First, we provide a theoretical framework for connecting graph summarization with change detection. Then, we empirically demonstrate its effectiveness on synthetic and real datasets.

Mingyi Li, Taira Tsuchiya, Kenji Yamanishi

This work studies online episodic tabular Markov decision processes (MDPs) with known transitions and develops best-of-both-worlds algorithms that achieve refined data-dependent regret bounds in the adversarial regime and variance-dependent regret bounds in the stochastic regime. We quantify MDP complexity using a first-order quantity and several new data-dependent measures for the adversarial regime, including a second-order quantity and a path-length measure, as well as variance-based measures for the stochastic regime. To adapt to these measures, we develop algorithms based on global optimization and policy optimization, both built on optimistic follow-the-regularized-leader with log-barrier regularization. For global optimization, our algorithms achieve first-order, second-order, and path-length regret bounds in the adversarial regime, and in the stochastic regime, they achieve a variance-aware gap-independent bound and a variance-aware gap-dependent bound that is polylogarithmic in the number of episodes. For policy optimization, our algorithms achieve the same data- and variance-dependent adaptivity, up to a factor of the episode horizon, by exploiting a new optimistic $Q$-function estimator. Finally, we establish regret lower bounds in terms of data-dependent complexity measures for the adversarial regime and a variance measure for the stochastic regime, implying that the regret upper bounds achieved by the global-optimization approach are nearly optimal.

Yuki Shibukawa, Taira Tsuchiya, Shinsaku Sakaue et al.

Online structured prediction is a task of sequentially predicting outputs with complex structures based on inputs and past observations, encompassing online classification. Recent studies showed that in the full-information setting, we can achieve finite bounds on the \textit{surrogate regret}, i.e. the extra target loss relative to the best possible surrogate loss. In practice, however, full-information feedback is often unrealistic as it requires immediate access to the whole structure of complex outputs. Motivated by this, we propose algorithms that work with less demanding feedback, bandit and delayed feedback. For bandit feedback, by using a standard inverse-weighted gradient estimator, we achieve a surrogate regret bound of $O(\sqrt{KT})$ for the time horizon $T$ and the size of the output set $K$. However, $K$ can be extremely large when outputs are highly complex, resulting in an undesirable bound. To address this issue, we propose another algorithm that achieves a surrogate regret bound of $O(T^{2/3})$, which is independent of $K$. This is achieved with a carefully designed pseudo-inverse matrix estimator. Furthermore, we numerically compare the performance of these algorithms, as well as existing ones. Regarding delayed feedback, we provide algorithms and regret analyses that cover various scenarios, including full-information and bandit feedback, as well as fixed and variable delays.

Kota Fukuzawa, Atsushi Suzuki, Kenji Yamanishi

In recent years, with the large-scale expansion of graph data, there has been an increased focus on Riemannian manifold data spaces other than Euclidean space. In particular, the development of hyperbolic spaces has been remarkable, and they have high expressive power for graph data with hierarchical structures. Normalized Maximum Likelihood (NML) is employed in regret minimization and model selection. However, existing formulations of NML have been developed primarily in Euclidean spaces and are inherently dependent on the choice of coordinate systems, making it non-trivial to extend NML to Riemannian manifolds. In this study, we define a new NML that reflects the geometric structure of Riemannian manifolds, called the Riemannian manifold NML (Rm-NML). This Rm-NML is invariant under coordinate transformations and coincides with the conventional NML under the natural parameterization in Euclidean space. We extend existing computational techniques for NML to the setting of Riemannian manifolds. Furthermore, we derive a method to simplify the computation of Rm-NML on Riemannian symmetric spaces, which encompass data spaces of growing interest such as hyperbolic spaces. To illustrate the practical application of our proposed method, we explicitly computed the Rm-NML for normal distributions on hyperbolic spaces.

Shintaro Fukushima, Kenji Yamanishi

This study addresses the issue of graph generation with generative models. In particular, we are concerned with graph community augmentation problem, which refers to the problem of generating unseen or unfamiliar graphs with a new community out of the probability distribution estimated with a given graph dataset. The graph community augmentation means that the generated graphs have a new community. There is a chance of discovering an unseen but important structure of graphs with a new community, for example, in a social network such as a purchaser network. Graph community augmentation may also be helpful for generalization of data mining models in a case where it is difficult to collect real graph data enough. In fact, there are many ways to generate a new community in an existing graph. It is desirable to discover a new graph with a new community beyond the given graph while we keep the structure of the original graphs to some extent for the generated graphs to be realistic. To this end, we propose an algorithm called the graph community augmentation (GCA). The key ideas of GCA are (i) to fit Gaussian mixture model (GMM) to data points in the latent space into which the nodes in the original graph are embedded, and (ii) to add data points in the new cluster in the latent space for generating a new community based on the minimum description length (MDL) principle. We empirically demonstrate the effectiveness of GCA for generating graphs with a new community structure on synthetic and real datasets.

Kento Urano, Ryo Yuki, Kenji Yamanishi

This paper proposes an early detection method for cluster structural changes. Cluster structure refers to discrete structural characteristics, such as the number of clusters, when data are represented using finite mixture models, such as Gaussian mixture models. We focused on scenarios in which the cluster structure gradually changed over time. For finite mixture models, the concept of mixture complexity (MC) measures the continuous cluster size by considering the cluster proportion bias and overlap between clusters. In this paper, we propose MC fusion as an extension of MC to handle situations in which multiple mixture numbers are possible in a finite mixture model. By incorporating the fusion of multiple models, our approach accurately captured the cluster structure during transitional periods of gradual change. Moreover, we introduce a method for detecting changes in the cluster structure by examining the transition of MC fusion. We demonstrate the effectiveness of our method through empirical analysis using both artificial and real-world datasets.

Atsushi Suzuki, Atsushi Nitanda, Taiji Suzuki et al.

Recent studies have experimentally shown that we can achieve in non-Euclidean metric space effective and efficient graph embedding, which aims to obtain the vertices' representations reflecting the graph's structure in the metric space. Specifically, graph embedding in hyperbolic space has experimentally succeeded in embedding graphs with hierarchical-tree structure, e.g., data in natural languages, social networks, and knowledge bases. However, recent theoretical analyses have shown a much higher upper bound on non-Euclidean graph embedding's generalization error than Euclidean one's, where a high generalization error indicates that the incompleteness and noise in the data can significantly damage learning performance. It implies that the existing bound cannot guarantee the success of graph embedding in non-Euclidean metric space in a practical training data size, which can prevent non-Euclidean graph embedding's application in real problems. This paper provides a novel upper bound of graph embedding's generalization error by evaluating the local Rademacher complexity of the model as a function set of the distances of representation couples. Our bound clarifies that the performance of graph embedding in non-Euclidean metric space, including hyperbolic space, is better than the existing upper bounds suggest. Specifically, our new upper bound is polynomial in the metric space's geometric radius $R$ and can be $O(\frac{1}{S})$ at the fastest, where $S$ is the training data size. Our bound is significantly tighter and faster than the existing one, which can be exponential to $R$ and $O(\frac{1}{\sqrt{S}})$ at the fastest. Specific calculations on example cases show that graph embedding in non-Euclidean metric space can outperform that in Euclidean space with much smaller training data than the existing bound has suggested.

Atsushi Suzuki, Atsushi Nitanda, Jing Wang et al.

Hyperbolic ordinal embedding (HOE) represents entities as points in hyperbolic space so that they agree as well as possible with given constraints in the form of entity i is more similar to entity j than to entity k. It has been experimentally shown that HOE can obtain representations of hierarchical data such as a knowledge base and a citation network effectively, owing to hyperbolic space's exponential growth property. However, its theoretical analysis has been limited to ideal noiseless settings, and its generalization error in compensation for hyperbolic space's exponential representation ability has not been guaranteed. The difficulty is that existing generalization error bound derivations for ordinal embedding based on the Gramian matrix do not work in HOE, since hyperbolic space is not inner-product space. In this paper, through our novel characterization of HOE with decomposed Lorentz Gramian matrices, we provide a generalization error bound of HOE for the first time, which is at most exponential with respect to the embedding space's radius. Our comparison between the bounds of HOE and Euclidean ordinal embedding shows that HOE's generalization error is reasonable as a cost for its exponential representation ability.

Shintaro Fukushima, Kenji Yamanishi

This paper addresses the issue of detecting hierarchical changes in latent variable models (HCDL) from data streams. There are three different levels of changes for latent variable models: 1) the first level is the change in data distribution for fixed latent variables, 2) the second one is that in the distribution over latent variables, and 3) the third one is that in the number of latent variables. It is important to detect these changes because we can analyze the causes of changes by identifying which level a change comes from (change interpretability). This paper proposes an information-theoretic framework for detecting changes of the three levels in a hierarchical way. The key idea to realize it is to employ the MDL (minimum description length) change statistics for measuring the degree of change, in combination with DNML (decomposed normalized maximum likelihood) code-length calculation. We give a theoretical basis for making reliable alarms for changes. Focusing on stochastic block models, we employ synthetic and benchmark datasets to empirically demonstrate the effectiveness of our framework in terms of change interpretability as well as change detection.

Pham Thuc Hung, Kenji Yamanishi

In this paper, we propose a novel information criteria-based approach to select the dimensionality of the word2vec Skip-gram (SG). From the perspective of the probability theory, SG is considered as an implicit probability distribution estimation under the assumption that there exists a true contextual distribution among words. Therefore, we apply information criteria with the aim of selecting the best dimensionality so that the corresponding model can be as close as possible to the true distribution. We examine the following information criteria for the dimensionality selection problem: the Akaike Information Criterion, Bayesian Information Criterion, and Sequential Normalized Maximum Likelihood (SNML) criterion. SNML is the total codelength required for the sequential encoding of a data sequence on the basis of the minimum description length. The proposed approach is applied to both the original SG model and the SG Negative Sampling model to clarify the idea of using information criteria. Additionally, as the original SNML suffers from computational disadvantages, we introduce novel heuristics for its efficient computation. Moreover, we empirically demonstrate that SNML outperforms both BIC and AIC. In comparison with other evaluation methods for word embedding, the dimensionality selected by SNML is significantly closer to the optimal dimensionality obtained by word analogy or word similarity tasks.

Linchuan Xu, Jun Huang, Atsushi Nitanda et al.

Spatial attention has been introduced to convolutional neural networks (CNNs) for improving both their performance and interpretability in visual tasks including image classification. The essence of the spatial attention is to learn a weight map which represents the relative importance of activations within the same layer or channel. All existing attention mechanisms are local attentions in the sense that weight maps are image-specific. However, in the medical field, there are cases that all the images should share the same weight map because the set of images record the same kind of symptom related to the same object and thereby share the same structural content. In this paper, we thus propose a novel global spatial attention mechanism in CNNs mainly for medical image classification. The global weight map is instantiated by a decision boundary between important pixels and unimportant pixels. And we propose to realize the decision boundary by a binary classifier in which the intensities of all images at a pixel are the features of the pixel. The binary classification is integrated into an image classification CNN and is to be optimized together with the CNN. Experiments on two medical image datasets and one facial expression dataset showed that with the proposed attention, not only the performance of four powerful CNNs which are GoogleNet, VGG, ResNet, and DenseNet can be improved, but also meaningful attended regions can be obtained, which is beneficial for understanding the content of images of a domain.

Shintaro Fukushima, Atsushi Nitanda, Kenji Yamanishi

In online learning from non-stationary data streams, it is necessary to learn robustly to outliers and to adapt quickly to changes in the underlying data generating mechanism. In this paper, we refer to the former attribute of online learning algorithms as robustness and to the latter as adaptivity. There is an obvious tradeoff between the two attributes. It is a fundamental issue to quantify and evaluate the tradeoff because it provides important information on the data generating mechanism. However, no previous work has considered the tradeoff quantitatively. We propose a novel algorithm called the stochastic approximation-based robustness-adaptivity algorithm (SRA) to evaluate the tradeoff. The key idea of SRA is to update parameters of distribution or sufficient statistics with the biased stochastic approximation scheme, while dropping data points with large values of the stochastic update. We address the relation between the two parameters: one is the step size of the stochastic approximation, and the other is the threshold parameter of the norm of the stochastic update. The former controls the adaptivity and the latter does the robustness. We give a theoretical analysis for the non-asymptotic convergence of SRA in the presence of outliers, which depends on both the step size and threshold parameter. Because SRA is formulated on the majorization-minimization principle, it is a general algorithm that includes many algorithms, such as the online EM algorithm and stochastic gradient descent. Empirical experiments for both synthetic and real datasets demonstrated that SRA was superior to previous methods.

Shunki Kyoya, Kenji Yamanishi

In model-based clustering using finite mixture models, it is a significant challenge to determine the number of clusters (cluster size). It used to be equal to the number of mixture components (mixture size); however, this may not be valid in the presence of overlaps or weight biases. In this study, we propose to continuously measure the cluster size in a mixture model by a new concept called mixture complexity (MC). It is formally defined from the viewpoint of information theory and can be seen as a natural extension of the cluster size considering overlap and weight bias. Subsequently, we apply MC to the issue of gradual clustering change detection. Conventionally, clustering changes has been considered to be abrupt, induced by the changes in the mixture size or cluster size. Meanwhile, we consider the clustering changes to be gradual in terms of MC; it has the benefits of finding the changes earlier and discerning the significant and insignificant changes. We further demonstrate that the MC can be decomposed according to the hierarchical structures of the mixture models; it helps us to analyze the detail of substructures.

Kenji Yamanishi

This paper introduces a new notion of dimensionality of probabilistic models from an information-theoretic view point. We call it the "descriptive dimension"(Ddim). We show that Ddim coincides with the number of independent parameters for the parametric class, and can further be extended to real-valued dimensionality when a number of models are mixed. The paper then derives the rate of convergence of the MDL (Minimum Description Length) learning algorithm which outputs a normalized maximum likelihood (NML) distribution with model of the shortest NML codelength. The paper proves that the rate is governed by Ddim. The paper also derives error probabilities of the MDL-based test for multiple model change detection. It proves that they are also governed by Ddim. Through the analysis, we demonstrate that Ddim is an intrinsic quantity which characterizes the performance of the MDL-based learning and change detection.

Kohei Miyaguchi, Kenji Yamanishi

We develop a new theoretical framework, the \emph{envelope complexity}, to analyze the minimax regret with logarithmic loss functions and derive a Bayesian predictor that adaptively achieves the minimax regret over high-dimensional $\ell_1$-balls within a factor of two. The prior is newly derived for achieving the minimax regret and called the \emph{spike-and-tails~(ST) prior} as it looks like. The resulting regret bound is so simple that it is completely determined with the smoothness of the loss function and the radius of the balls except with logarithmic factors, and it has a generalized form of existing regret/risk bounds. In the preliminary experiment, we confirm that the ST prior outperforms the conventional minimax-regret prior under non-high-dimensional asymptotics.

Yosuke Enokida, Atsushi Suzuki, Kenji Yamanishi

A hyperbolic space has been shown to be more capable of modeling complex networks than a Euclidean space. This paper proposes an explicit update rule along geodesics in a hyperbolic space. The convergence of our algorithm is theoretically guaranteed, and the convergence rate is better than the conventional Euclidean gradient descent algorithm. Moreover, our algorithm avoids the "bias" problem of existing methods using the Riemannian gradient. Experimental results demonstrate the good performance of our algorithm in the \Poincare embeddings of knowledge base data.

Kohei Miyaguchi, Kenji Yamanishi

We tackle the problem of penalty selection of regularization on the basis of the minimum description length (MDL) principle. In particular, we consider that the design space of the penalty function is high-dimensional. In this situation, the luckiness-normalized-maximum-likelihood(LNML)-minimization approach is favorable, because LNML quantifies the goodness of regularized models with any forms of penalty functions in view of the minimum description length principle, and guides us to a good penalty function through the high-dimensional space. However, the minimization of LNML entails two major challenges: 1) the computation of the normalizing factor of LNML and 2) its minimization in high-dimensional spaces. In this paper, we present a novel regularization selection method (MDL-RS), in which a tight upper bound of LNML (uLNML) is minimized with local convergence guarantee. Our main contribution is the derivation of uLNML, which is a uniform-gap upper bound of LNML in an analytic expression. This solves the above challenges in an approximate manner because it allows us to accurately approximate LNML and then efficiently minimize it. The experimental results show that MDL-RS improves the generalization performance of regularized estimates specifically when the model has redundant parameters.

Taito Lee, Shin Matsushima, Kenji Yamanishi

This paper introduces the combinatorial Boolean model (CBM), which is defined as the class of linear combinations of conjunctions of Boolean attributes. This paper addresses the issue of learning CBM from labeled data. CBM is of high knowledge interpretability but naïve learning of it requires exponentially large computation time with respect to data dimension and sample size. To overcome this computational difficulty, we propose an algorithm GRAB (GRAfting for Boolean datasets), which efficiently learns CBM within the $L_1$-regularized loss minimization framework. The key idea of GRAB is to reduce the loss minimization problem to the weighted frequent itemset mining, in which frequent patterns are efficiently computable. We employ benchmark datasets to empirically demonstrate that GRAB is effective in terms of computational efficiency, prediction accuracy and knowledge discovery.

Motohide Higaki, Kai Morino, Hiroshi Murata et al.

This study addresses the issue of predicting the glaucomatous visual field loss from patient disease datasets. Our goal is to accurately predict the progress of the disease in individual patients. As very few measurements are available for each patient, it is difficult to produce good predictors for individuals. A recently proposed clustering-based method enhances the power of prediction using patient data with similar spatiotemporal patterns. Each patient is categorized into a cluster of patients, and a predictive model is constructed using all of the data in the class. Predictions are highly dependent on the quality of clustering, but it is difficult to identify the best clustering method. Thus, we propose a method for aggregating cluster-based predictors to obtain better prediction accuracy than from a single cluster-based prediction. Further, the method shows very high performances by hierarchically aggregating experts generated from several cluster-based methods. We use real datasets to demonstrate that our method performs significantly better than conventional clustering-based and patient-wise regression methods, because the hierarchical aggregating strategy has a mechanism whereby good predictors in a small community can thrive.

So Hirai, Kenji Yamanishi

We are concerned with the issue of how to calculate the normalized maximum likelihood (NML) code-length. There is a problem that the normalization term of the NML code-length may diverge when it is continuous and unbounded and a straightforward computation of it is highly expensive when the data domain is finite . In previous works it has been investigated how to calculate the NML code-length for specific types of distributions. We first propose a general method for computing the NML code-length for the exponential family. Then we specifically focus on Gaussian mixture model (GMM), and propose a new efficient method for computing the NML to them. We develop it by generalizing Rissanen's re-normalizing technique. Then we apply this method to the clustering issue, in which a clustering structure is modeled using a GMM, and the main task is to estimate the optimal number of clusters on the basis of the NML code-length. We demonstrate using artificial data sets the superiority of the NML-based clustering over other criteria such as AIC, BIC in terms of the data size required for high accuracy rate to be achieved.