arXiv subjects

Alessandro Rinaldo

Publications and source records attributed to Alessandro Rinaldo.

At least 73 records · Page 4Linked to original sources

arXiv2019

Minimax Rates for Estimating the Dimension of a Manifold

Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be estimated. We characterize the statistical difficulty of this problem by deriving upper and lower bounds on the minimax rate for estimating the dimension. First, we consider the problem of testing the hypothesis that the support of the data-generating probability distribution is a well-behaved manifold of intrinsic dimension $d_1$ versus the alternative that it is of dimension $d_2$, with $d_{1}<d_{2}$. With an i.i.d. sample of size $n$, we provide an upper bound on the probability of choosing the wrong dimension of $O\left( n^{-\left(d_{2}/d_{1}-1-ε\right)n} \right)$, where $ε$ is an arbitrarily small positive number. The proof is based on bounding the length of the traveling salesman path through the data points. We also demonstrate a lower bound of $Ω\left( n^{-(2d_{2}-2d_{1}+ε)n} \right)$, by applying Le Cam's lemma with a specific set of $d_{1}$-dimensional probability distributions. We then extend these results to get minimax rates for estimating the dimension of well-behaved manifolds. We obtain an upper bound of order $O \left( n^{-(\frac{1}{m-1}-ε)n} \right)$ and a lower bound of order $Ω\left( n^{-(2+ε)n} \right)$, where $m$ is the embedding dimension.

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arXiv2019

DBSCAN: Optimal Rates For Density Based Clustering

We study the problem of optimal estimation of the density cluster tree under various assumptions on the underlying density. Building up from the seminal work of Chaudhuri et al. [2014], we formulate a new notion of clustering consistency which is better suited to smooth densities, and derive minimax rates of consistency for cluster tree estimation for Holder smooth densities of arbitrary degree α. We present a computationally efficient, rate optimal cluster tree estimator based on a straightforward extension of the popular density-based clustering algorithm DBSCAN by Ester et al. [1996]. The procedure relies on a kernel density estimator with an appropriate choice of the kernel and bandwidth to produce a sequence of nested random geometric graphs whose connected components form a hierarchy of clusters. The resulting optimal rates for cluster tree estimation depend on the degree of smoothness of the underlying density and, interestingly, match minimax rates for density estimation under the supremum norm. Our results complement and extend the analysis of the DBSCAN algorithm in Sriperumbudur and Steinwart [2012]. Finally, we consider level set estimation and cluster consistency for densities with jump discontinuities, where the sizes of the jumps and the distance among clusters are allowed to vanish as the sample size increases. We demonstrate that our DBSCAN-based algorithm remains minimax rate optimal in this setting as well.

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arXiv2019

Cryptocurrency Price Prediction and Trading Strategies Using Support Vector Machines

Few assets in financial history have been as notoriously volatile as cryptocurrencies. While the long term outlook for this asset class remains unclear, we are successful in making short term price predictions for several major crypto assets. Using historical data from July 2015 to November 2019, we develop a large number of technical indicators to capture patterns in the cryptocurrency market. We then test various classification methods to forecast short-term future price movements based on these indicators. On both PPV and NPV metrics, our classifiers do well in identifying up and down market moves over the next 1 hour. Beyond evaluating classification accuracy, we also develop a strategy for translating 1-hour-ahead class predictions into trading decisions, along with a backtester that simulates trading in a realistic environment. We find that support vector machines yield the most profitable trading strategies, which outperform the market on average for Bitcoin, Ethereum and Litecoin over the past 22 months, since January 2018.

q-fin.TR↗

arXiv2019

Predictive clustering

We show how to convert any clustering into a prediction set. This has the effect of converting the clustering into a (possibly overlapping) union of spheres or ellipsoids. The tuning parameters can be chosen to minimize the size of the prediction set. When applied to k-means clustering, this method solves several problems: the method tells us how to choose k, how to merge clusters and how to replace the Voronoi partition with more natural shapes. We show that the same reasoning can be applied to other clustering methods.

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arXiv2019

Hierarchical Models for Independence Structures of Networks

We introduce a new family of network models, called hierarchical network models, that allow us to represent in an explicit manner the stochastic dependence among the dyads (random ties) of the network. In particular, each member of this family can be associated with a graphical model defining conditional independence clauses among the dyads of the network, called the dependency graph. Every network model with dyadic independence assumption can be generalized to construct members of this new family. Using this new framework, we generalize the Erdös-Rényi and beta-models to create hierarchical Erdös-Rényi and beta-models. We describe various methods for parameter estimation as well as simulation studies for models with sparse dependency graphs.

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arXiv2019

Are sample means in multi-armed bandits positively or negatively biased?

It is well known that in stochastic multi-armed bandits (MAB), the sample mean of an arm is typically not an unbiased estimator of its true mean. In this paper, we decouple three different sources of this selection bias: adaptive \emph{sampling} of arms, adaptive \emph{stopping} of the experiment, and adaptively \emph{choosing} which arm to study. Through a new notion called ``optimism'' that captures certain natural monotonic behaviors of algorithms, we provide a clean and unified analysis of how optimistic rules affect the sign of the bias. The main takeaway message is that optimistic sampling induces a negative bias, but optimistic stopping and optimistic choosing both induce a positive bias. These results are derived in a general stochastic MAB setup that is entirely agnostic to the final aim of the experiment (regret minimization or best-arm identification or anything else). We provide examples of optimistic rules of each type, demonstrate that simulations confirm our theoretical predictions, and pose some natural but hard open problems.

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arXiv2019

Statistical Analysis of Nearest Neighbor Methods for Anomaly Detection

Nearest-neighbor (NN) procedures are well studied and widely used in both supervised and unsupervised learning problems. In this paper we are concerned with investigating the performance of NN-based methods for anomaly detection. We first show through extensive simulations that NN methods compare favorably to some of the other state-of-the-art algorithms for anomaly detection based on a set of benchmark synthetic datasets. We further consider the performance of NN methods on real datasets, and relate it to the dimensionality of the problem. Next, we analyze the theoretical properties of NN-methods for anomaly detection by studying a more general quantity called distance-to-measure (DTM), originally developed in the literature on robust geometric and topological inference. We provide finite-sample uniform guarantees for the empirical DTM and use them to derive misclassification rates for anomalous observations under various settings. In our analysis we rely on Huber's contamination model and formulate mild geometric regularity assumptions on the underlying distribution of the data.

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arXiv2019

Univariate Mean Change Point Detection: Penalization, CUSUM and Optimality

The problem of univariate mean change point detection and localization based on a sequence of $n$ independent observations with piecewise constant means has been intensively studied for more than half century, and serves as a blueprint for change point problems in more complex settings. We provide a complete characterization of this classical problem in a general framework in which the upper bound $σ^2$ on the noise variance, the minimal spacing $Δ$ between two consecutive change points and the minimal magnitude $κ$ of the changes, are allowed to vary with $n$. We first show that consistent localization of the change points, when the signal-to-noise ratio $\frac{κ\sqrtΔ}σ < \sqrt{\log(n)}$, is impossible. In contrast, when $\frac{κ\sqrtΔ}σ$ diverges with $n$ at the rate of at least $\sqrt{\log(n)}$, we demonstrate that two computationally-efficient change point estimators, one based on the solution to an $\ell_0$-penalized least squares problem and the other on the popular wild binary segmentation algorithm, are both consistent and achieve a localization rate of the order $\frac{σ^2}{κ^2} \log(n)$. We further show that such rate is minimax optimal, up to a $\log(n)$ term.

math.ST↗

arXiv2019

Time Series Featurization via Topological Data Analysis

We develop a novel algorithm for feature extraction in time series data by leveraging tools from topological data analysis. Our algorithm provides a simple, efficient way to successfully harness topological features of the attractor of the underlying dynamical system for an observed time series. The proposed methodology relies on the persistent landscapes and silhouette of the Rips complex obtained after a de-noising step based on principal components applied to a time-delayed embedding of a noisy, discrete time series sample. We analyze the stability properties of the proposed approach and show that the resulting TDA-based features are robust to sampling noise. Experiments on synthetic and real-world data demonstrate the effectiveness of our approach. We expect our method to provide new insights on feature extraction from granular, noisy time series data.

cs.CG↗

arXiv2019

Optimal nonparametric change point detection and localization

We study change point detection and localization for univariate data in fully nonparametric settings in which, at each time point, we acquire an i.i.d. sample from an unknown distribution. We quantify the magnitude of the distributional changes at the change points using the Kolmogorov--Smirnov distance. We allow all the relevant parameters -- the minimal spacing between two consecutive change points, the minimal magnitude of the changes in the Kolmogorov--Smirnov distance, and the number of sample points collected at each time point -- to change with the length of time series. We generalize the renowned binary segmentation (e.g. Scott and Knott, 1974) algorithm and its variant, the wild binary segmentation of Fryzlewicz (2014), both originally designed for univariate mean change point detection problems, to our nonparametric settings and exhibit rates of consistency for both of them. In particular, we prove that the procedure based on wild binary segmentation is nearly minimax rate-optimal. We further demonstrate a phase transition in the space of model parameters that separates parameter combinations for which consistent localization is possible from the ones for which this task is statistical unfeasible. Finally, we provide extensive numerical experiments to support our theory. R code is available at https://github.com/hernanmp/NWBS.

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arXiv2019

Estimating the Reach of a Manifold

Various problems in manifold estimation make use of a quantity called the reach, denoted by $τ\_M$, which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we study the geometry of the reach through an approximation perspective. We derive new geometric results on the reach for submanifolds without boundary. An estimator $\hatτ$ of $τ\_{M}$ is proposed in a framework where tangent spaces are known, and bounds assessing its efficiency are derived. In the case of i.i.d. random point cloud $\mathbb{X}\_{n}$, $\hatτ(\mathbb{X}\_{n})$ is showed to achieve uniform expected loss bounds over a $\mathcal{C}^3$-like model. Finally, we obtain upper and lower bounds on the minimax rate for estimating the reach.

math.ST↗

arXiv2019

Markov Properties of Discrete Determinantal Point Processes

Determinantal point processes (DPPs) are probabilistic models for repulsion. When used to represent the occurrence of random subsets of a finite base set, DPPs allow to model global negative associations in a mathematically elegant and direct way. Discrete DPPs have become popular and computationally tractable models for solving several machine learning tasks that require the selection of diverse objects, and have been successfully applied in numerous real-life problems. Despite their popularity, the statistical properties of such models have not been adequately explored. In this note, we derive the Markov properties of discrete DPPs and show how they can be expressed using graphical models.

math.ST↗

arXiv2018

On Exchangeability in Network Models

We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size.

math.ST↗

arXiv2018

Optimal Covariance Change Point Localization in High Dimension

We study the problem of change point detection for covariance matrices in high dimensions. We assume that we observe a sequence {X_i}_{i=1,...,n} of independent and centered p-dimensional sub-Gaussian random vectors whose covariance matrices are piecewise constant. Our task is to recover with high accuracy the number and locations of the change points, which are assumed unknown. Our generic model setting allows for all the model parameters to change with n, including the dimension p, the minimal spacing between consecutive change points, the magnitude of smallest change size and the maximal Orlicz- 2 norm of the covariance matrices of the sample points. Without assuming any additional structural assumption, such as low rank matrices or having sparse principle components, we set up a general framework and a benchmark result for the covariance change point detection problem. We introduce two procedures, one based on the binary segmentation algorithm (e.g. Vostrikova, 1981) and the other on its extension known as wild binary segmentation of Fryzlewicz (2014), and demonstrate that, under suitable conditions, both procedures are able to consistently es- timate the number and locations of change points. Our second algorithm, called Wild Binary Segmentation through Independent Projection (WBSIP), is shown to be optimal in the sense of allowing for the minimax scaling in all the relevant parameters. Our minimax analysis reveals a phase transition effect based on the problem of change point localization. To the best of our knowledge, this type of results has not been established elsewhere in the high-dimensional change point detection literature.

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arXiv2018

Bootstrapping and Sample Splitting For High-Dimensional, Assumption-Free Inference

Several new methods have been proposed for performing valid inference after model selection. An older method is sampling splitting: use part of the data for model selection and part for inference. In this paper we revisit sample splitting combined with the bootstrap (or the Normal approximation). We show that this leads to a simple, assumption-free approach to inference and we establish results on the accuracy of the method. In fact, we find new bounds on the accuracy of the bootstrap and the Normal approximation for general nonlinear parameters with increasing dimension which we then use to assess the accuracy of regression inference. We show that an alternative, called the image bootstrap, has higher coverage accuracy at the cost of more computation. We define new parameters that measure variable importance and that can be inferred with greater accuracy than the usual regression coefficients. There is a inference-prediction tradeoff: splitting increases the accuracy and robustness of inference but can decrease the accuracy of the predictions.

math.ST↗

arXiv2017

Random Networks, Graphical Models, and Exchangeability

We study conditional independence relationships for random networks and their interplay with exchangeability. We show that, for finitely exchangeable network models, the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts. We then characterize all possible Markov structures for finitely exchangeable random graphs, thereby identifying a new class of Markov network models corresponding to bidirected Kneser graphs. In particular, we demonstrate that the fundamental property of dissociatedness corresponds to a Markov property for exchangeable networks described by bidirected line graphs. Finally we study those exchangeable models that are also summarized in the sense that the probability of a network only depends onthe degree distribution, and identify a class of models that is dual to the Markov graphs of Frank and Strauss (1986). Particular emphasis is placed on studying consistency properties of network models under the process of forming subnetworks and we show that the only consistent systems of Markov properties correspond to the empty graph, the bidirected line graph of the complete graph, and the complete graph.

math.ST↗

arXiv2017

Uniform Asymptotic Inference and the Bootstrap After Model Selection

Recently, Tibshirani et al. (2016) proposed a method for making inferences about parameters defined by model selection, in a typical regression setting with normally distributed errors. Here, we study the large sample properties of this method, without assuming normality. We prove that the test statistic of Tibshirani et al. (2016) is asymptotically valid, as the number of samples n grows and the dimension d of the regression problem stays fixed. Our asymptotic result holds uniformly over a wide class of nonnormal error distributions. We also propose an efficient bootstrap version of this test that is provably (asymptotically) conservative, and in practice, often delivers shorter intervals than those from the original normality-based approach. Finally, we prove that the test statistic of Tibshirani et al. (2016) does not enjoy uniform validity in a high-dimensional setting, when the dimension d is allowed grow.

math.ST↗

arXiv2017

Distribution-Free Predictive Inference For Regression

We develop a general framework for distribution-free predictive inference in regression, using conformal inference. The proposed methodology allows for the construction of a prediction band for the response variable using any estimator of the regression function. The resulting prediction band preserves the consistency properties of the original estimator under standard assumptions, while guaranteeing finite-sample marginal coverage even when these assumptions do not hold. We analyze and compare, both empirically and theoretically, the two major variants of our conformal framework: full conformal inference and split conformal inference, along with a related jackknife method. These methods offer different tradeoffs between statistical accuracy (length of resulting prediction intervals) and computational efficiency. As extensions, we develop a method for constructing valid in-sample prediction intervals called {\it rank-one-out} conformal inference, which has essentially the same computational efficiency as split conformal inference. We also describe an extension of our procedures for producing prediction bands with locally varying length, in order to adapt to heteroskedascity in the data. Finally, we propose a model-free notion of variable importance, called {\it leave-one-covariate-out} or LOCO inference. Accompanying this paper is an R package {\tt conformalInference} that implements all of the proposals we have introduced. In the spirit of reproducibility, all of our empirical results can also be easily (re)generated using this package.

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