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In recent years, large language models (LLMs) have demonstrated exceptional power in various domains, including information retrieval. Most of the previous practices involve leveraging these models to create a single embedding for each query, each passage, or each document individually, a strategy exemplified and used by the Retrieval-Augmented Generation (RAG) framework. While this method has proven effective, we argue that it falls short in fully capturing the nuanced intricacies of document-level texts due to its reliance on a relatively coarse-grained representation. To address this limitation, we introduce a novel, fine-grained approach aimed at enhancing the accuracy of relevance scoring for long documents. Our methodology firstly segments a long document into blocks, each of which is embedded using an LLM, for matching with the query representation. When calculating the relevance score, we aggregate the query-block relevance scores through a weighted sum method, yielding a comprehensive score for the query with the entire document. Despite its apparent simplicity, our experimental findings reveal that this approach outperforms standard representation methods and achieves a significant reduction in embedding generation latency. Moreover, by carefully optimizing pairwise loss functions, superior performances have been achieved.

Understanding deep models is crucial for deploying them in safety-critical applications. We introduce GIFT, a framework for deriving post-hoc, global, interpretable, and faithful textual explanations for vision classifiers. GIFT starts from local faithful visual counterfactual explanations and employs (vision) language models to translate those into global textual explanations. Crucially, GIFT provides a verification stage measuring the causal effect of the proposed explanations on the classifier decision. Through experiments across diverse datasets, including CLEVR, CelebA, and BDD, we demonstrate that GIFT effectively reveals meaningful insights, uncovering tasks, concepts, and biases used by deep vision classifiers. The framework is released at https://github.com/valeoai/GIFT.

The rapid development of large language models (LLMs) like Llama has significantly advanced information retrieval (IR) systems. However, using LLMs for long documents, as in RankLLaMA, remains challenging due to computational complexity, especially concerning input token length. Furthermore, the internal mechanisms of LLMs during ranking are still not fully understood. In this paper, we first explore the internal workings of LLMs during relevance judgement and identify that specific attention heads play a crucial role in aligning relevant tokens. This observation inspires us to revisit the block pre-ranking strategy used in KeyB, which remains state-of-the-art (SOTA) on the TREC 2019 DL document ranking dataset. Building on these insights, we develop KeyB2, an advanced long document IR approach that integrates block pre-ranking with the performance of LLMs. KeyB2 efficiently identifies and processes the most relevant blocks, reducing computational costs and improving ranking effectiveness. Additionally, we introduce a new bi-encoder block matching strategy for KeyB2. Comprehensive experiments on long-document datasets, including TREC 2019 DL, Robust04, and MLDR-zh, show that KeyB2 outperforms baselines like RankLLaMA and KeyB by reducing reranking time and GPU memory usage while enhancing retrieval performance, achieving new SOTA results on TREC 2019 DL with higher NDCG@10 and MAP scores.

We study the boundary behavior of the Kobayashi-Royden metric and the Kobayashi hyperbolicity of domains in Riemannian manifolds. As an application, we prove a Fatou type theorem on the existence, almost everywhere, of non tangential limits for bounded conformal harmonic immersed discs., Comment: 12 pages

Tree-based ensemble methods such as random forests, gradient-boosted trees, and Bayesianadditive regression trees have been successfully used for regression problems in many applicationsand research studies. In this paper, we study ensemble versions of probabilisticregression trees that provide smooth approximations of the objective function by assigningeach observation to each region with respect to a probability distribution. We prove thatthe ensemble versions of probabilistic regression trees considered are consistent, and experimentallystudy their bias-variance trade-off and compare them with the state-of-the-art interms of performance prediction.

Recent studies have demonstrated that the ability of dense retrieval models to generalize to target domains with different distributions is limited, which contrasts with the results obtained with interaction-based models. Prior attempts to mitigate this challenge involved leveraging adversarial learning and query generation approaches, but both approaches nevertheless resulted in limited improvements. In this paper, we propose to combine the query-generation approach with a self-supervision approach in which pseudo-relevance labels are automatically generated on the target domain. To accomplish this, a T5-3B model is utilized for pseudo-positive labeling, and meticulous hard negatives are chosen. We also apply this strategy on conversational dense retrieval model for conversational search. A similar pseudo-labeling approach is used, but with the addition of a query-rewriting module to rewrite conversational queries for subsequent labeling. This proposed approach enables a model's domain adaptation with real queries and documents from the target dataset. Experiments on standard dense retrieval and conversational dense retrieval models both demonstrate improvements on baseline models when they are fine-tuned on the pseudo-relevance labeled data., Comment: 12 pages, accepted by COLING 2024

This paper introduces a new structural causal model tailored for representing threshold-based IT systems and presents a new algorithm designed to rapidly detect root causes of anomalies in such systems. When root causes are not causally related, the method is proven to be correct; while an extension is proposed based on the intervention of an agent to relax this assumption. Our algorithm and its agent-based extension leverage causal discovery from offline data and engage in subgraph traversal when encountering new anomalies in online data. Our extensive experiments demonstrate the superior performance of our methods, even when applied to data generated from alternative structural causal models or real IT monitoring data., Comment: Accepted to CIKM 2024

We establish a smoothness result for families of biholomorphisms between smooth families of strongly pseudoconvex domains, each with trivial biholomorphism group. This is accomplished by considering the Riemannian geometry of their Bergman metrics and proving a result about the smoothness of families of isometries between smooth families of Riemannian manifolds., Comment: v1: 31 pages. Comments welcome!

We give an example of a parabolic holomorphic self-map $f$ of the unit ball $\mathbb B^2\subset \mathbb C^2$ whose canonical Kobayashi hyperbolic semi-model is given by an elliptic automorphism of the disc $\mathbb D\subset \mathbb C$, which can be chosen to be different from the identity. As a consequence, in contrast to the one dimensional case, this provides a first example of a holomorphic self-map of the unit ball which has points with zero hyperbolic step and points with nonzero hyperbolic step, solving an open question and showing that parabolic dynamics in the ball $\\mathbb B^2$ is radically different from parabolic dynamics in the disc. The example is obtained via a geometric method, embedding the ball $\mathbb B^2$ as a domain $\Omega$ in the bidisc $\\mathbb D\times \mathbb{H}$ that is forward invariant and absorbing for the map $(z,w)\mapsto (e^{i\theta}z,w+1)$, where $\mathbb H\subset \mathbb C$ denotes the right half-plane. We also show that a complete Kobayashi hyperbolic domain $\Omega$ with such properties cannot be Gromov hyperbolic w.r.t. the Kobayashi distance (hence, it cannot be biholomorphic to $\\mathbb B^2$) if an additional quantitative geometric condition is satisfied., Comment: Final version, accepted to J. Geom. Anal

We study the problem of identifiability of the total effect of an intervention from observational time series in the situation, common in practice, where one only has access to abstractions of the true causal graph. We consider here two abstractions: the extended summary causal graph, which conflates all lagged causal relations but distinguishes between lagged and instantaneous relations, and the summary causal graph which does not give any indication about the lag between causal relations. We show that the total effect is always identifiable in extended summary causal graphs and provide sufficient conditions for identifiability in summary causal graphs. We furthermore provide adjustment sets allowing to estimate the total effect whenever it is identifiable.

Information technology (IT) systems are vital for modern businesses, handling data storage, communication, and process automation. Monitoring these systems is crucial for their proper functioning and efficiency, as it allows collecting extensive observational time series data for analysis. The interest in causal discovery is growing in IT monitoring systems as knowing causal relations between different components of the IT system helps in reducing downtime, enhancing system performance and identifying root causes of anomalies and incidents. It also allows proactive prediction of future issues through historical data analysis. Despite its potential benefits, applying causal discovery algorithms on IT monitoring data poses challenges, due to the complexity of the data. For instance, IT monitoring data often contains misaligned time series, sleeping time series, timestamp errors and missing values. This paper presents case studies on applying causal discovery algorithms to different IT monitoring datasets, highlighting benefits and ongoing challenges., Comment: Accepted to the UAI 2023 Workshop on The History and Development of Search Methods for Causal Structure

We introduce an analog of the Kobayashi-Royden metric on a Riemannian manifold and study its basic properties., Comment: 11 pp

Constraint-based methods and noise-based methods are two distinct families of methods proposed for uncovering causal graphs from observational data. However, both operate under strong assumptions that may be challenging to validate or could be violated in real-world scenarios. In response to these challenges, there is a growing interest in hybrid methods that amalgamate principles from both methods, showing robustness to assumption violations. This paper introduces a novel comprehensive framework for hybridizing constraint-based and noise-based methods designed to uncover causal graphs from observational time series. The framework is structured into two classes. The first class employs a noise-based strategy to identify a super graph, containing the true graph, followed by a constraint-based strategy to eliminate unnecessary edges. In the second class, a constraint-based strategy is applied to identify a skeleton, which is then oriented using a noise-based strategy. The paper provides theoretical guarantees for each class under the condition that all assumptions are satisfied, and it outlines some properties when assumptions are violated. To validate the efficacy of the framework, two algorithms from each class are experimentally tested on simulated data, realistic ecological data, and real datasets sourced from diverse applications. Notably, two novel datasets related to Information Technology monitoring are introduced within the set of considered real datasets. The experimental results underscore the robustness and effectiveness of the hybrid approaches across a broad spectrum of datasets., Comment: Accepted in TMLR: https://openreview.net/forum?id=PGLbZpVk2n

Although neural information retrieval has witnessed great improvements, recent works showed that the generalization ability of dense retrieval models on target domains with different distributions is limited, which contrasts with the results obtained with interaction-based models. To address this issue, researchers have resorted to adversarial learning and query generation approaches; both approaches nevertheless resulted in limited improvements. In this paper, we propose to use a self-supervision approach in which pseudo-relevance labels are automatically generated on the target domain. To do so, we first use the standard BM25 model on the target domain to obtain a first ranking of documents, and then use the interaction-based model T53B to re-rank top documents. We further combine this approach with knowledge distillation relying on an interaction-based teacher model trained on the source domain. Our experiments reveal that pseudo-relevance labeling using T53B and the MiniLM teacher performs on average better than other approaches and helps improve the state-of-the-art query generation approach GPL when it is fine-tuned on the pseudo-relevance labeled data., Comment: 16 pages

We present different constructions of abstract boundaries for bounded complete (Kobayashi) hyperbolic domains in ${\mathbb C}^d$, $d \geq 1$. These constructions essentially come from the geometric theory of metric spaces. We also present, as an application, some extension results concerning biholomorphic maps.

This study addresses the problem of learning an extended summary causal graph on time series. The algorithms we propose fit within the well-known constraint-based framework for causal discovery and make use of information-theoretic measures to determine (in)dependencies between time series. We first introduce generalizations of the causation entropy measure to any lagged or instantaneous relations, prior to using this measure to construct extended summary causal graphs by adapting two well-known algorithms, namely PC and FCI. The behavior of our methods is illustrated through several experiments run on simulated and real datasets.

We consider in this paper the problem of predicting the ability of a startup to attract investments using freely, publicly available data. Information about startups on the web usually comes either as unstructured data from news, social networks, and websites or as structured data from commercial databases, such as Crunchbase. The possibility of predicting the success of a startup from structured databases has been studied in the literature and it has been shown that initial public offerings (IPOs), mergers and acquisitions (M\&A) as well as funding events can be predicted with various machine learning techniques. In such studies, heterogeneous information from the web and social networks is usually used as a complement to the information coming from databases. However, building and maintaining such databases demands tremendous human effort. We thus study here whether one can solely rely on readily available sources of information, such as the website of a startup, its social media activity as well as its presence on the web, to predict its funding events. As illustrated in our experiments, the method we propose yields results comparable to the ones making also use of structured data available in private databases.

We introduce the notion of locally visible and locally Gromov hyperbolic domains in $\mathbb C^d$. We prove that a bounded domain in $\mathbb C^d$ is locally visible and locally Gromov hyperbolic if and only if it is (globally) visible and Gromov hyperbolic with respect to the Kobayashi distance. This allows to detect, from local information near the boundary, those domains which are Gromov hyperbolic and for which biholomorphisms extend continuously up to the boundary., Comment: 25 pages

In this paper we prove a metric version of Hartogs' theorem where the holomorphic function is replaced by a locally symmetric Hermitian metric. As an application, we prove that if the Kobayashi metric on a strongly pseudoconvex domain with $\mathcal{C}^2$ smooth boundary is a K\"ahler metric, then the universal cover of the domain is the unit ball., Comment: 16 pages. v2: substantial changes (new and improved results, new title). Comments welcome

On a wide range of natural language processing and information retrieval tasks, transformer-based models, particularly pre-trained language models like BERT, have demonstrated tremendous effectiveness. Due to the quadratic complexity of the self-attention mechanism, however, such models have difficulties processing long documents. Recent works dealing with this issue include truncating long documents, in which case one loses potential relevant information, segmenting them into several passages, which may lead to miss some information and high computational complexity when the number of passages is large, or modifying the self-attention mechanism to make it sparser as in sparse-attention models, at the risk again of missing some information. We follow here a slightly different approach in which one first selects key blocks of a long document by local query-block pre-ranking, and then few blocks are aggregated to form a short document that can be processed by a model such as BERT. Experiments conducted on standard Information Retrieval datasets demonstrate the effectiveness of the proposed approach., Comment: 34 pages, accepted by ACM Transactions on Information Systems (TOIS)

This study addresses the problem of learning a summary causal graph on time series with potentially different sampling rates. To do so, we first propose a new causal temporal mutual information measure for time series. We then show how this measure relates to an entropy reduction principle that can be seen as a special case of the probability raising principle. We finally combine these two ingredients in PC-like and FCI-like algorithms to construct the summary causal graph. There algorithm are evaluated on several datasets, which shows both their efficacy and efficiency., Comment: Published in Entropy (2022)

Information retrieval (IR) systems traditionally aim to maximize metrics built on rankings, such as precision or NDCG. However, the non-differentiability of the ranking operation prevents direct optimization of such metrics in state-of-the-art neural IR models, which rely entirely on the ability to compute meaningful gradients. To address this shortcoming, we propose SmoothI, a smooth approximation of rank indicators that serves as a basic building block to devise differentiable approximations of IR metrics. We further provide theoretical guarantees on SmoothI and derived approximations, showing in particular that the approximation errors decrease exponentially with an inverse temperature-like hyperparameter that controls the quality of the approximations. Extensive experiments conducted on four standard learning-to-rank datasets validate the efficacy of the listwise losses based on SmoothI, in comparison to previously proposed ones. Additional experiments with a vanilla BERT ranking model on a text-based IR task also confirm the benefits of our listwise approach.

The dominant approaches to text representation in natural language rely on learning embeddings on massive corpora which have convenient properties such as compositionality and distance preservation. In this paper, we develop a novel method to learn a heavy-tailed embedding with desirable regularity properties regarding the distributional tails, which allows to analyze the points far away from the distribution bulk using the framework of multivariate extreme value theory. In particular, a classifier dedicated to the tails of the proposed embedding is obtained which performance outperforms the baseline. This classifier exhibits a scale invariance property which we leverage by introducing a novel text generation method for label preserving dataset augmentation. Numerical experiments on synthetic and real text data demonstrate the relevance of the proposed framework and confirm that this method generates meaningful sentences with controllable attribute, e.g. positive or negative sentiment.

Metric learning has been successful in learning new metrics adapted to numerical datasets. However, its development on categorical data still needs further exploration. In this paper, we propose a method, called CPML for \emph{categorical projected metric learning}, that tries to efficiently~(i.e. less computational time and better prediction accuracy) address the problem of metric learning in categorical data. We make use of the Value Distance Metric to represent our data and propose new distances based on this representation. We then show how to efficiently learn new metrics. We also generalize several previous regularizers through the Schatten $p$-norm and provides a generalization bound for it that complements the standard generalization bound for metric learning. Experimental results show that our method provides

We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space $\mathbb C^n$ to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in $\mathbb C^3$ that is Kobayashi hyperbolic and has a nonempty core. In particular, this domain is not biholomorphic to a bounded domain in $\mathbb C^3$ and the mentioned above sufficient condition for Kobayashi hyperbolicity is not necessary., Comment: 18 pages, 3 figures

We propose in this paper a new, hybrid document embedding approach in order to address the problem of document similarities with respect to the technical content. To do so, we employ a state-of-the-art graph techniques to first extract the keyphrases (composite keywords) of documents and, then, use them to score the sentences. Using the ranked sentences, we propose two approaches to embed documents and show their performances with respect to two baselines. With domain expert annotations, we illustrate that the proposed methods can find more relevant documents and outperform the baselines up to 27% in terms of NDCG.

This paper describes our submission to the E2E NLG Challenge. Recently, neural seq2seq approaches have become mainstream in NLG, often resorting to pre- (respectively post-) processing delexicalization (relexicalization) steps at the word-level to handle rare words. By contrast, we train a simple character level seq2seq model, which requires no pre/post-processing (delexicalization, tokenization or even lowercasing), with surprisingly good results. For further improvement, we explore two re-ranking approaches for scoring candidates. We also introduce a synthetic dataset creation procedure, which opens up a new way of creating artificial datasets for Natural Language Generation.

Tree-based ensemble methods, as Random Forests and Gradient Boosted Trees, have been successfully used for regression in many applications and research studies. Furthermore, these methods have been extended in order to deal with uncertainty in the output variable, using for example a quantile loss in Random Forests (Meinshausen, 2006). To the best of our knowledge, no extension has been provided yet for dealing with uncertainties in the input variables, even though such uncertainties are common in practical situations. We propose here such an extension by showing how standard regression trees optimizing a quadratic loss can be adapted and learned while taking into account the uncertainties in the inputs. By doing so, one no longer assumes that an observation lies into a single region of the regression tree, but rather that it belongs to each region with a certain probability. Experiments conducted on several data sets illustrate the good behavior of the proposed extension., Comment: 9 pages

We study how the existence of a negatively pinched K\"ahler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete K\"ahler metric, with pinched negative holomorphic bisectional curvature outside a compact set, then the boundary of the domain does not contain any complex subvariety of positive domain. Moreover, if the boundary of the domain is smooth, then it is of finite type in the sense of D'Angelo. We also use curvature to provide a characterization of strong pseudoconvexity amongst convex domains. In particular, we show that a convex domain with $C^{2,\alpha}$ boundary is strongly pseudoconvex if and only if it admits a complete K\"ahler metric with sufficiently tight pinched negative holomorphic sectional curvature outside a compact set., Comment: 26 pages

Let $(\phi_t)$ be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let $\Omega$ be the starlike at infinity domain image of the Koenigs function of $(\phi_t)$. In this paper we completely characterize the type of convergence of the orbits of $(\phi_t)$ to the Denjoy-Wolff point in terms of the shape of $\Omega$. In particular we prove that the convergence is non-tangential if and only if the domain $\Omega$ is `quasi-symmetric with respect to vertical axes'. We also prove that such conditions are equivalent to the curve $[0,\infty)\ni t\mapsto \phi_t(z)$ being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of $\Omega$., Comment: 25 pages, 3 figures

We study the homeomorphic extension of biholomorphisms between convex domains in $\mathbb C^d$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov boundary and the topological boundaries of the domains and the dynamical properties of commuting 1-Lipschitz maps in Gromov hyperbolic spaces. This approach not only allows us to prove extensions for biholomorphisms, but for more general quasi-isometries between the domains endowed with their Kobayashi distances., Comment: 27 pages; final version, to appear Math. Ann

We study in this paper the problem of jointly clustering and learning representations. As several previous studies have shown, learning representations that are both faithful to the data to be clustered and adapted to the clustering algorithm can lead to better clustering performance, all the more so that the two tasks are performed jointly. We propose here such an approach for $k$-Means clustering based on a continuous reparametrization of the objective function that leads to a truly joint solution. The behavior of our approach is illustrated on various datasets showing its efficacy in learning representations for objects while clustering them., Comment: Under consideration at Pattern Recognition Letters

Let $\mathbb D$ be the unit disc in $\mathbb C$ and let $f:\mathbb D \to \mathbb C$ be a Riemann map, $\Delta=f(\mathbb D)$. We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a compactly divergent sequence $\{z_n\}\subset \Delta$ has the property that $\{f^{-1}(z_n)\}$ converges orthogonally to a point of $\partial \mathbb D$. We also give some applications of this to the slope problem for continuous semigroups of holomorphic self-maps of $\mathbb D$., Comment: 15 pages

We study the backward invariant set of one-parameter semigroups of holomorphic self-maps of the unit disc. Such a set is foliated in maximal invariant curves and its open connected components are petals, which are, in fact, images of Poggi-Corradini's type pre-models. Hyperbolic petals are in one-to-one correspondence with repelling fixed points, while only parabolic semigroups can have parabolic petals. Petals have locally connected boundaries and, except a very particular case, they are indeed Jordan domains. The boundary of a petal contains the Denjoy-Wolff point and, except such a fixed point, the closure of a petal contains either no other boundary fixed point or a unique repelling fixed point. We also describe petals in terms of geometric and analytic behavior of K\"onigs functions using divergence rate and universality of models. Moreover, we construct a semigroup having a repelling fixed point in such a way that the intertwining map of the pre-model is not regular., Comment: 35 pages

Let $\Delta\subsetneq \mathbb C$ be a simply connected domain, let $f:\mathbb D \to \Delta$ be a Riemann map and let $\{z_k\}\subset \Delta$ be a compactly divergent sequence. Using Gromov's hyperbolicity theory, we show that $\{f^{-1}(z_k)\}$ converges non-tangentially to a point of $\partial \mathbb D$ if and only if there exists a simply connected domain $U\subsetneq \mathbb C$ such that $\Delta\subset U$ and $\Delta$ contains a tubular hyperbolic neighborhood of a geodesic of $U$ and $\{z_k\}$ is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that if $(\phi_t)$ is a non-elliptic semigroup of holomorphic self-maps of $\mathbb D$ with K\"onigs function $h$ and $h(\mathbb D)$ contains a vertical Euclidean sector, then $\phi_t(z)$ converges to the Denjoy-Wolff point non-tangentially for every $z\in \mathbb D$ as $t\to +\infty$. Using new localization results for the hyperbolic distance, we also construct an example of a parabolic semigroup which converges non-tangentially to the Denjoy-Wolff point but oscillating, in the sense that the slope of the trajectories is not a single point., Comment: 32 pages; 1 figure - final version, to appear in Trans. Amer. Math. Soc

We prove that two smooth families of 2-connected domains in $\cc$ are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct, for $m \geq 3$, two smooth families of smoothly bounded $m$-connected domains in $\cc$, and for $n\geq2$, two families of strictly pseudoconvex domains in $\cc^n$, that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains.

Latent factor models are increasingly popular for modeling multi-relational knowledge graphs. By their vectorial nature, it is not only hard to interpret why this class of models works so well, but also to understand where they fail and how they might be improved. We conduct an experimental survey of state-of-the-art models, not towards a purely comparative end, but as a means to get insight about their inductive abilities. To assess the strengths and weaknesses of each model, we create simple tasks that exhibit first, atomic properties of binary relations, and then, common inter-relational inference through synthetic genealogies. Based on these experimental results, we propose new research directions to improve on existing models., Comment: 30+3 pages, submitted to the Journal of Artificial Intelligence Research

We give a description of complex geodesics and we study the structure of stationary discs in some non-convex domains for which complex geodesics are not unique., Comment: 9 pages

We propose here an extended attention model for sequence-to-sequence recurrent neural networks (RNNs) designed to capture (pseudo-)periods in time series. This extended attention model can be deployed on top of any RNN and is shown to yield state-of-the-art performance for time series forecasting on several univariate and multivariate time series.

In statistical relational learning, knowledge graph completion deals with automatically understanding the structure of large knowledge graphs---labeled directed graphs---and predicting missing relationships---labeled edges. State-of-the-art embedding models propose different trade-offs between modeling expressiveness, and time and space complexity. We reconcile both expressiveness and complexity through the use of complex-valued embeddings and explore the link between such complex-valued embeddings and unitary diagonalization. We corroborate our approach theoretically and show that all real square matrices---thus all possible relation/adjacency matrices---are the real part of some unitarily diagonalizable matrix. This results opens the door to a lot of other applications of square matrices factorization. Our approach based on complex embeddings is arguably simple, as it only involves a Hermitian dot product, the complex counterpart of the standard dot product between real vectors, whereas other methods resort to more and more complicated composition functions to increase their expressiveness. The proposed complex embeddings are scalable to large data sets as it remains linear in both space and time, while consistently outperforming alternative approaches on standard link prediction benchmarks., Comment: 38 pages, accepted in JMLR. This is an extended version of the article "Complex embeddings for simple link prediction" (ICML 2016)

We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let $F:\mathbb B^n\to \mathbb C^n$ be a univalent map from the unit ball whose image $D$ is convex. Let $\mathcal S\subset \partial \mathbb B^n$ be the set of points $\xi$ such that $\lim_{z\to \xi}\|F(z)\|=\infty$. Then we prove that $\mathcal S$ is either empty, or contains one or two points and $F$ extends as a homeomorphism $\tilde{F}:\overline{\mathbb B^n}\setminus \mathcal S\to \overline{D}$. Moreover, $\mathcal S=\emptyset$ if $D$ is bounded, $\mathcal S$ has one point if $D$ has one connected component at $\infty$ and $\mathcal S$ has two points if $D$ has two connected components at $\infty$ and, up to composition with an affine map, $F$ is an extension of the strip map in the plane to higher dimension., Comment: last version, accepted in Math. Ann

We show that the graph of a holomorphic motion of the unit disc cannot be biholomorphic to a strongly pseudoconvex domain in C n .

Multivariate time series naturally exist in many fields, like energy, bioinformatics, signal processing, and finance. Most of these applications need to be able to compare these structured data. In this context, dynamic time warping (DTW) is probably the most common comparison measure. However, not much research effort has been put into improving it by learning. In this paper, we propose a novel method for learning similarities based on DTW, in order to improve time series classification. Making use of the uniform stability framework, we provide the first theoretical guarantees in the form of a generalization bound for linear classification. The experimental study shows that the proposed approach is efficient, while yielding sparse classifiers., Comment: Techreport

In statistical relational learning, the link prediction problem is key to automatically understand the structure of large knowledge bases. As in previous studies, we propose to solve this problem through latent factorization. However, here we make use of complex valued embeddings. The composition of complex embeddings can handle a large variety of binary relations, among them symmetric and antisymmetric relations. Compared to state-of-the-art models such as Neural Tensor Network and Holographic Embeddings, our approach based on complex embeddings is arguably simpler, as it only uses the Hermitian dot product, the complex counterpart of the standard dot product between real vectors. Our approach is scalable to large datasets as it remains linear in both space and time, while consistently outperforming alternative approaches on standard link prediction benchmarks., Comment: 10+2 pages, accepted at ICML 2016

Diagonalization, or eigenvalue decomposition, is very useful in many areas of applied mathematics, including signal processing and quantum physics. Matrix decomposition is also a useful tool for approximating matrices as the product of a matrix and its transpose, which relates to unitary diagonalization. As stated by the spectral theorem, only normal matrices are unitarily diagonalizable. However we show that all real square matrices are the real part of some unitarily diagonalizable matrix.

We introduce a prime end-type theory on complete Kobayashi hyperbolic manifolds using horosphere sequences. This allows to introduce a new notion of boundary-new even in the unit disc in the complex space-the horosphere boundary, and a topology on the manifold together with its horosphere boundary, the horosphere topology. We prove that a bounded strongly pseudoconvex domain endowed with the horosphere topology is homeomorphic to its Euclidean closure, while for the polydisc such a horosphere topology is not even Hausdorff and is different from the Gromov topology. We use this theory to study boundary behavior of univalent maps from bounded strongly pseudoconvex domains. Among other things, we prove that every univalent map of the unit ball whose image is bounded and convex, extends as a homeomorphism up to the closure. Such a result, relying in an essential way on our theory and on the Gromov hyperbolicity theory, is completely new, dealing with non smooth domains., Comment: This is the final version of the paper