We consider the mixed regression problem with two components, under adversarial and stochastic noise. We give a convex optimization formulation that provably recovers the true solution, and provide upper bounds on the recovery errors for both arbitrary noise and stochastic noise settings. We also give matching minimax lower bounds (up to log factors), showing that under certain assumptions, our algorithm is information-theoretically optimal. Our results represent the first tractable algorithm guaranteeing successful recovery with tight bounds on recovery errors and sample complexity.
In this work we consider the setting with two mixture components. Many interesting applications have binary latent factors: gene mutation present not, gender, healthy sick individual, children adult, etc.; see also the examples in @cite_25 . Theoretically, the minimax rate was previously unknown even in the two-component case. Extension to more than two components is of great interest.
@cite_25
2017565019 2769906263 2749155890 2059638643
This paper considers a new variant of the two-dimensional bin packing problem where each rectangle is assigned a due date and each bin has a fixed processing time. Hence the objective is not only to minimize the number of bins, but also to minimize the maximum lateness of the rectangles. This problem is motivated by the cutting of stock sheets and the potential increased efficiency that might be gained by drawing on a larger pool of demand pieces by mixing orders, while also aiming to ensure a certain level of customer service. We propose a genetic algorithm for searching the solution space, which uses a new placement heuristic for decoding the gene based on the best fit heuristic designed for the strip packing problems. The genetic algorithm employs an innovative crossover operator that considers several different children from each pair of parents. Further, the dual objective is optimized hierarchically with the primary objective periodically alternating between maximum lateness and number of bins. As a result, the approach produces several non-dominated solutions with different trade-offs. Two further approaches are implemented. One is based on a previous Unified Tabu Search, suitably modified to tackle this revised problem. The other is randomized descent and serves as a benchmark for comparing the results. Comprehensive computational results are presented, which show that the Unified Tabu Search still works well in minimizing the bins, but the genetic algorithm performs slightly better. When also considering maximum lateness, the genetic algorithm is considerably better. In this paper we study the asymptotic properties of Bayesian multiple testing procedures for a large class of Gaussian scale mixture pri- ors. We study two types of multiple testing risks: a Bayesian risk proposed in (2011) where the data are assume to come from a mixture of normal, and a frequentist risk similar to the one proposed by Arias-Castro and Chen (2017). Following the work of van der (2016), we give general conditions on the prior such that both risks can be bounded. For the Bayesian risk, the bound is almost sharp. This result show that under these conditions, the considered class of continuous prior can be competitive with the usual two-group model (e.g. spike and slab priors). We also show that if the non-zeros component of the parameter are large enough, the minimax risk can be made asymptotically null. The separation rates obtained are consistent with the one that could be guessed from the existing literature (see van der , 2017b). For both problems, we then give conditions under which an adaptive version of the result can be obtained. We propose a novel latent factor model for recommending product size fits Small, Fit, Large to customers. Latent factors for customers and products in our model correspond to their physical true size, and are learnt from past product purchase and returns data. The outcome for a customer, product pair is predicted based on the difference between customer and product true sizes, and efficient algorithms are proposed for computing customer and product true size values that minimize two loss function variants. In experiments with Amazon shoe datasets, we show that our latent factor models incorporating personas, and leveraging return codes show a 17-21 AUC improvement compared to baselines. In an online A B test, our algorithms show an improvement of 0.49 in percentage of Fit transactions over control. In this paper, we present a method that enable both homology-based approach and composition-based approach to further study the functional core (i.e., microbial core and gene core, correspondingly). In the proposed method, the identification of major functionality groups is achieved by generative topic modeling, which is able to extract useful information from unlabeled data. We first show that generative topic model can be used to model the taxon abundance information obtained by homology-based approach and study the microbial core. The model considers each sample as a "document,” which has a mixture of functional groups, while each functional group (also known as a "latent topic”) is a weight mixture of species. Therefore, estimating the generative topic model for taxon abundance data will uncover the distribution over latent functions (latent topic) in each sample. Second, we show that, generative topic model can also be used to study the genome-level composition of "N-mer” features (DNA subreads obtained by composition-based approaches). The model consider each genome as a mixture of latten genetic patterns (latent topics), while each functional pattern is a weighted mixture of the "N-mer” features, thus the existence of core genomes can be indicated by a set of common N-mer features. After studying the mutual information between latent topics and gene regions, we provide an explanation of the functional roles of uncovered latten genetic patterns. The experimental results demonstrate the effectiveness of proposed method.