Tensor decomposition has emerged as a prominent technique to learn low-dimensional representation under the supervision of reconstruction error, primarily benefiting data inference tasks like completion and imputation, but not classification task. We argue that the non-uniqueness and rotation invariance of tensor decomposition allow us to identify the directions with largest class-variability and simple graph Laplacian can effectively achieve this objective. Therefore we propose a novel Pseudo Laplacian Contrast (PLC) tensor decomposition framework, which integrates the data augmentation and cross-view Laplacian to enable the extraction of class-aware representations while effectively capturing the intrinsic low-rank structure within reconstruction constraint. An unsupervised alternative optimization algorithm is further developed to iteratively estimate the pseudo graph and minimize the loss using Alternating Least Square (ALS). Extensive experimental results on various datasets demonstrate the effectiveness of our approach.

When applying nonnegative matrix factorization (NMF), generally the rank parameter is unknown. Such rank in NMF, called the nonnegative rank, is usually estimated heuristically since computing the exact value of it is NP-hard. In this work, we propose an approximation method to estimate such rank while solving NMF on-the-fly. We use sum-of-norm (SON), a group-lasso structure that encourages pairwise similarity, to reduce the rank of a factor matrix where the rank is overestimated at the beginning. On various datasets, SON-NMF is able to reveal the correct nonnegative rank of the data without any prior knowledge nor tuning. SON-NMF is a nonconvx nonsmmoth non-separable non-proximable problem, solving it is nontrivial. First, as rank estimation in NMF is NP-hard, the proposed approach does not enjoy a lower computational complexity. Using a graph-theoretic argument, we prove that the complexity of the SON-NMF is almost irreducible. Second, the per-iteration cost of any algorithm solving SON-NMF is possibly high, which motivated us to propose a first-order BCD algorithm to approximately solve SON-NMF with a low per-iteration cost, in which we do so by the proximal average operator. Lastly, we propose a simple greedy method for post-processing. SON-NMF exhibits favourable features for applications. Beside the ability to automatically estimate the rank from data, SON-NMF can deal with rank-deficient data matrix, can detect weak component with small energy. Furthermore, on the application of hyperspectral imaging, SON-NMF handle the issue of spectral variability naturally.

We study optimal data pooling for shared learning in two common maintenance operations: condition-based maintenance and spare parts management. We consider a set of systems subject to Poisson input -- the degradation or demand process -- that are coupled through an a-priori unknown rate. Decision problems involving these systems are high-dimensional Markov decision processes (MDPs) and hence notoriously difficult to solve. We present a decomposition result that reduces such an MDP to two-dimensional MDPs, enabling structural analyses and computations. Leveraging this decomposition, we (i) demonstrate that pooling data can lead to significant cost reductions compared to not pooling, and (ii) show that the optimal policy for the condition-based maintenance problem is a control limit policy, while for the spare parts management problem, it is an order-up-to level policy, both dependent on the pooled data.

Seasonal-trend decomposition is one of the most fundamental concepts in time series analysis that supports various downstream tasks, including time series anomaly detection and forecasting. However, existing decomposition methods rely on batch processing with a time complexity of O(W), where W is the number of data points within a time window. Therefore, they cannot always efficiently support real-time analysis that demands low processing delay. To address this challenge, we propose OneShotSTL, an efficient and accurate algorithm that can decompose time series online with an update time complexity of O(1). OneShotSTL is more than $1,000$ times faster than the batch methods, with accuracy comparable to the best counterparts. Extensive experiments on real-world benchmark datasets for downstream time series anomaly detection and forecasting tasks demonstrate that OneShotSTL is from 10 to over 1,000 times faster than the state-of-the-art methods, while still providing comparable or even better accuracy.

This paper proposes a simple approach to derive efficient error bounds for learning multiple components with sparsity-inducing regularization. Indeed, we show that for such regularization schemes, known decompositions of the Rademacher complexity over the components can be used in a more efficient manner to result in tighter bounds without too much effort. We give examples of application to switching regression and center-based clustering/vector quantization. Then, the complete workflow is illustrated on the problem of subspace clustering, for which decomposition results were not previously available. For all these problems, the proposed approach yields risk bounds with mild dependencies on the number of components and completely removes this dependency for nonconvex regularization schemes that could not be handled by previous methods.

The inadequacy of traditional forecasting model based on EEMD in practical work. For WTI, the first four IMFs decomposed by EEMD are suitable as inputs. Considering the actual demand of crude oil price forecasting, a novel model based on ensemble empirical mode decomposition (EEMD) and long short-term memory (LSTM) is proposed. In practical work, the model trained by historical data will be used in later data. Then the forecasting models based on EEMD need re-execute EEMD to update decomposition results of price series after getting new data.