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Phylogenetic trees constitute an interesting class of objects for stochastic processes due to the non-standard nature of the space they inhabit. In particular, many statistical applications require the construction of Markov processes on the space of trees, whose cardinality grows superexponentially with the number of leaves considered. We investigate whether certain lower-dimensional projections of tree space preserve the Markov property in tree-valued Markov processes. We study exact lumpability of tree shapes and $\varepsilon$-lumpability of clades, exploiting the combinatorial structure of the SPR graph to obtain bounds on the lumping error under the random walk and Metropolis-Hastings processes. Finally, we show how to use these results in empirical investigation, leveraging exact and $\varepsilon$-lumpability to improve Monte Carlo estimation of tree-related quantities.

The evaluation of recommendation systems is a complex task. The offline and online evaluation metrics for recommender systems are ambiguous in their true objectives. The majority of recently published papers benchmark their methods using ill-posed offline evaluation methodology that often fails to predict true online performance. Because of this, the impact that academic research has on the industry is reduced. The aim of our research is to investigate and compare the online performance of offline evaluation metrics. We show that penalizing popular items and considering the time of transactions during the evaluation significantly improves our ability to choose the best recommendation model for a live recommender system. Our results, averaged over five large-size real-world live data procured from recommenders, aim to help the academic community to understand better offline evaluation and optimization criteria that are more relevant for real applications of recommender systems., Comment: Accepted to evalRS 2023@KDD

We study a variant of the Generalized Excited Random Walk (GERW) on $\mathbb{Z}^d$ introduced by Menshikov, Popov, Ram\'irez and Vachkovskaia in [Ann. Probab. 40 (5), 2012]. It consists in a particular version of the model studied in [arXiv preprint arXiv:2211.05715, 2022] where excitation may or may not occur according to a time-dependent probability. Specifically, given $\{p_n\}_{n \ge 1}$, $p_n \in (0, 1]$ for all $n \ge 1$, whenever the process visits a site at time $n$ for the first time, with probability $p_n$ it gains a drift in a determined direction. Otherwise, it behaves as a $d$-martingale with zero-mean vector. We refer to the model as a GERW in time-inhomogeneous Bernoulli environment, in short, $p_n$-GERW. Under bounded jumps hypothesis and with $p_n \approx n^{-\beta}$, $\beta \geq 1/2$, we show a series of results for the $p_n$-GERW depending on the value of $\beta$ and on the dimension. Specifically, for $\beta > 1/2$, $d\geq 2$ and $\beta=1/2$ and $d=2$ we obtain a Functional Central Limit Theorem. Finally, for $\beta=1/2$ and $d \ge 22$ we obtain that the diffusively rescaled $p_n$-GERW is tight, and every limit point $\mathcal{Y}$ satisfies $W_t \cdot \ell + c_1 \sqrt{t} \preceq \mathcal{Y}_t \cdot \ell \preceq W_t \cdot \ell + c_2 \sqrt{t}$, where $c_1$ and $c_2$ are known positive constants, $W_{\cdot}$ is a Brownian motion and $\ell$ is the direction of the drift., Comment: Add a new author. We solved conjecture 2.2 for d=>22. We removed restrictions on the direction in Theorem 2.3 and solved for d => 22. We made a series of adjustments and corrections

We study inductive matrix completion (matrix completion with side information) under an i.i.d. subgaussian noise assumption at a low noise regime, with uniform sampling of the entries. We obtain for the first time generalization bounds with the following three properties: (1) they scale like the standard deviation of the noise and in particular approach zero in the exact recovery case; (2) even in the presence of noise, they converge to zero when the sample size approaches infinity; and (3) for a fixed dimension of the side information, they only have a logarithmic dependence on the size of the matrix. Differently from many works in approximate recovery, we present results both for bounded Lipschitz losses and for the absolute loss, with the latter relying on Talagrand-type inequalities. The proofs create a bridge between two approaches to the theoretical analysis of matrix completion, since they consist in a combination of techniques from both the exact recovery literature and the approximate recovery literature., Comment: 30 Pages, 1 figure; Accepted for publication at AAAI 2023

We consider a variation of the Generalized Excited Random Walk (GERW) in dimension $d\ge 2$ where the lower bound on the drift for excited jumps is time-dependent and decays to zero. We show that if the lower bound decays slower that $n^{-\beta}$ ($n$ is time), for $\beta$ depending on the transitions of the process, the GERW is transient in the direction of the drift., Comment: arXiv admin note: text overlap with arXiv:2201.10666 Add some comments and remarks for a better understanding, besides correcting some typos

We study an extension of the generalized excited random walk (GERW) on $\mathbb{Z}^d$ introduced in [Ann. Probab. 40 (5), 2012, [7]] by Menshikov, Popov, Ram\'irez and Vachkovskaia. Our extension consists in studying a version of the GERW where excitation depends on a random environment. Given $p \in (0,1]$ (a parameter of the model) whenever the process visits a site for the first time, with probability $p$ it gains a drift in a given direction (could be any direction of the unit sphere). Otherwise, with probability $1-p$, it behaves as a $d$-martingale with zero-mean vector. Whenever the process visits an already-visited site, the process acts again as a $d$-martingale with zero-mean vector. We refer to the model as a GERW in Bernoulli environment, in short $p$-GERW. Under the same hypothesis of [7] (bounded jumps, uniform ellipticity), we show that the $p$-GERW is ballistic for all $p\in (0,1]$. Under the stronger assumptions that the increments of the regeneration times associated to the $p$-GERW are i.i.d. (condition which is satisfied, for example, for the excited random walk in a Bernoulli i.i.d. environment), we also obtain a Law of Large Numbers and a Central Limit Theorem., Comment: The results of the paper are now contemplated in other works and we prefer to stop its disclosure

We propose orthogonal inductive matrix completion (OMIC), an interpretable approach to matrix completion based on a sum of multiple orthonormal side information terms, together with nuclear-norm regularization. The approach allows us to inject prior knowledge about the singular vectors of the ground truth matrix. We optimize the approach by a provably converging algorithm, which optimizes all components of the model simultaneously. We study the generalization capabilities of our method in both the distribution-free setting and in the case where the sampling distribution admits uniform marginals, yielding learning guarantees that improve with the quality of the injected knowledge in both cases. As particular cases of our framework, we present models which can incorporate user and item biases or community information in a joint and additive fashion. We analyse the performance of OMIC on several synthetic and real datasets. On synthetic datasets with a sliding scale of user bias relevance, we show that OMIC better adapts to different regimes than other methods. On real-life datasets containing user/items recommendations and relevant side information, we find that OMIC surpasses the state-of-the-art, with the added benefit of greater interpretability., Comment: To appear in Transactions of Neural Networks and Learning Systems (TNNLS)

Activity coefficients, which are a measure of the non-ideality of liquid mixtures, are a key property in chemical engineering with relevance to modeling chemical and phase equilibria as well as transport processes. Although experimental data on thousands of binary mixtures are available, prediction methods are needed to calculate the activity coefficients in many relevant mixtures that have not been explored to-date. In this report, we propose a probabilistic matrix factorization model for predicting the activity coefficients in arbitrary binary mixtures. Although no physical descriptors for the considered components were used, our method outperforms the state-of-the-art method that has been refined over three decades while requiring much less training effort. This opens perspectives to novel methods for predicting physico-chemical properties of binary mixtures with the potential to revolutionize modeling and simulation in chemical engineering., Comment: Published version: J. Phys. Chem. Lett. 11 (2020) 981-985; https://pubs.acs.org/doi/full/10.1021/acs.jpclett.9b03657