# sample complexity

• Le Song and Animashree Anandkumar and Bo Dai and Bo Xie

### Nonparametric Estimation of Multi-View Latent Variable Models (pdf)

Spectral methods have greatly advanced the estimation of latent variable models, generating a sequence of novel and efficient algorithms with strong theoretical guarantees. However, current spectral algorithms are largely restricted to mixtures of discrete or Gaussian distributions. In this paper, we propose a kernel method for learning multi-view latent variable models, allowing each mixture component to be nonparametric and learned from data in an unsupervised fashion. The key idea of our method is to embed the joint distribution of a multi-view latent variable model into a reproducing kernel Hilbert space, and then the latent parameters are recovered using a robust tensor power method. We establish that the sample complexity for the proposed method is quadratic in the number of latent components and is a low order polynomial in the other relevant parameters. Thus, our nonparametric tensor approach to learning latent variable models enjoys good sample and computational efficiencies. As a special case of our framework, we also obtain a first unsupervised conditional density estimator of the kind with provable guarantees. In both synthetic and real world datasets, the nonparametric tensor power method compares favorably to EM algorithm and other spectral algorithms.

• Hadi Daneshmand and Manuel Gomez-Rodriguez and Le Song and Bernhard Schoelkopf

### Estimating Diffusion Network Structures: Recovery Conditions, Sample Complexity & Soft-thresholding Algorithm (pdf)

Information spreads across social and technological networks, but often the network structures are hidden from us and we only observe the traces left by the diffusion processes, called cascades. Can we recover the hidden network structures from these observed cascades? What kind of cascades and how many cascades do we need? Are there some network structures which are more difficult than others to recover? Can we design efficient inference algorithms with provable guarantees? Despite the increasing availability of cascade data and methods for inferring networks from these data, a thorough theoretical understanding of the above questions remains largely unexplored in the literature. In this paper, we investigate the network structure inference problem for a general family of continuous-time diffusion models using an l1-regularized likelihood maximization framework. We show that, as long as the cascade sampling process satisfies a natural incoherence condition, our framework can recover the correct network structure with high probability if we observe O(d^3 log N) cascades, where d is the maximum number of parents of a node and N is the total number of nodes. Moreover, we develop a simple and efficient soft-thresholding inference algorithm, which we use to illustrate the consequences of our theoretical results, and show that our framework outperforms other alternatives in practice.

• Cun Mu and Bo Huang and John Wright and Donald Goldfarb

Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms (SNN) of the unfolding matrices of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a $K$-way $n$$\times$$n$$\times$$\cdots$$\times n$ tensor of Tucker rank $(r, r, \ldots, r)$ from Gaussian measurements requires $\Omega( r n^{K-1 • Liping Liu and Thomas Dietterich ### Learnability of the Superset Label Learning Problem (pdf) In the Superset Label Learning (SLL) problem, weak supervision is provided in the form of a {\it superset • Daniel Hsu and Sivan Sabato ### Heavy-tailed regression with a generalized median-of-means (pdf) This work proposes a simple and computationally efficient estimator for linear regression, and other smooth and strongly convex loss minimization problems. We prove loss approximation guarantees that hold for general distributions, including those with heavy tails. All prior results only hold for estimators which either assume bounded or subgaussian distributions, require prior knowledge of distributional properties, or are not known to be computationally tractable. In the special case of linear regression with possibly heavy-tailed responses and with bounded and well-conditioned covariates in$d$-dimensions, we show that a random sample of size$\tilde{O

• Nan Du and Yingyu Liang and Maria Balcan and Le Song

### Influence Function Learning in Information Diffusion Networks (pdf)

Can we learn the influence of a set of people in a social network from cascades of information diffusion? This question is often addressed by a two-stage approach: first learn a diffusion model, and then calculate the influence based on the learned model. Thus, the success of this approach relies heavily on the correctness of the diffusion model which is hard to verify for real world data. In this paper, we exploit the insight that the influence functions in many diffusion models are coverage functions, and propose a novel parameterization of such functions using a convex combination of random basis functions. Moreover, we propose an efficient maximum likelihood based algorithm to learn such functions directly from cascade data, and hence bypass the need to specify a particular diffusion model in advance. We provide both theoretical and empirical analysis for our approach, showing that the proposed approach can provably learn the influence function with low sample complexity, be robust to the unknown diffusion models, and significantly outperform existing approaches in both synthetic and real world data.

• Robert Busa-Fekete and Eyke Huellermeier and Balázs Szörényi

### Preference-Based Rank Elicitation using Statistical Models: The Case of Mallows (pdf)

We address the problem of rank elicitation assuming that the underlying data generating process is characterized by a probability distribution on the set of all rankings (total orders) of a given set of items. Instead of asking for complete rankings, however, our learner is only allowed to query pairwise preferences. Using information of that kind, the goal of the learner is to reliably predict properties of the distribution, such as the most probable top-item, the most probable ranking, or the distribution itself. More specifically, learning is done in an online manner, and the goal is to minimize sample complexity while guaranteeing a certain level of confidence.

• Jinfeng Yi and Lijun Zhang and Jun Wang and Rong Jin and Anil Jain

### A Single-Pass Algorithm for Efficiently Recovering Sparse Cluster Centers of High-dimensional Data (pdf)

Learning a statistical model for high-dimensional data is an important topic in machine learning. Although this problem has been well studied in the supervised setting, little is known about its unsupervised counterpart. In this work, we focus on the problem of clustering high-dimensional data with sparse centers. In particular, we address the following open question in unsupervised learning: is it possible to reliably cluster high-dimensional data when the number of samples is smaller than the data dimensionality?" We develop an efficient clustering algorithm that is able to estimate sparse cluster centers with a single pass over the data. Our theoretical analysis shows that the proposed algorithm is able to accurately recover cluster centers with only $O(s\log d)$ number of samples (data points), provided all the cluster centers are $s$-sparse vectors in a $d$ dimensional space. Experimental results verify both the effectiveness and efficiency of the proposed clustering algorithm compared to the state-of-the-art algorithms on several benchmark datasets.

• Sanjeev Arora and Aditya Bhaskara and Rong Ge and Tengyu Ma

We give algorithms with provable guarantees that learn a class of deep nets in the generative model view popularized by Hinton and others. Our generative model is an n node multilayer neural net that has degree at most $n^{\gamma • Rashish Tandon and Pradeep Ravikumar ### Learning Graphs with a Few Hubs (pdf) We consider the problem of recovering the graph structure of a hub-networked'' Ising model given iid samples, under high-dimensional settings, where number of nodes$p$could be potentially larger than the number of samples$n\$. By a hub-networked'' graph, we mean a graph with a few hub nodes'' with very large degrees. State of the art estimators for Ising models have a sample complexity that scales polynomially with the maximum node-degree, and are thus ill-suited to recovering such graphs with a few hub nodes. Some recent proposals for specifically recovering hub graphical models do not come with theoretical guarantees, and even empirically provide limited improvements over vanilla Ising model estimators. Here, we show that under such low sample settings, instead of estimating difficult'' components such as hub-neighborhoods, we can use quantitative indicators of our inability to do so, and thereby identify hub-nodes. This simple procedure allows us to recover hub-networked graphs with very strong statistical guarantees even under very low sample settings.

2013-2014 ICML | International Conference on Machine Learning