Usingalgebraic geometry,weestablish that UPCA is a well-defined algebraic problem in the sense that the only matrices of minimal rank that agree with the given data are row-permutations of the ground-truth matrix, arising as the unique solutions of a polynomial system of equations.

Suppose certain data points are overly contaminated, then the existing principal component analysis (PCA) methods are frequently incapable of filtering out and eliminating the excessively polluted ones, which potentially lead to the functional degeneration of the corresponding models. To tackle the issue, we propose a general framework namely robust weight learning with adaptive neighbors (RWL-AN), via which adaptive weight vector is automatically obtained with both robustness and sparse neighbors. More significantly, the degree of the sparsity is steerable such that only exact k well-fitting samples with least reconstruction errors are activated during the optimization, while the residual samples, i.e., the extreme noised ones are eliminated for the global robustness. Additionally, the framework is further applied to PCA problem to demonstrate the superiority and effectiveness of the proposed RWL-AN model.

In reality, the presence of outliers in data largely reduces the performance of PCA approaches.

Update: Thanks for the feedback and I have read them. Yet I don't think it has convinced me to change my decision. For Q2, if the framework is general, the authors should have extended it more than one case. Otherwise, the authors should focus on PCA instead of claiming the framework to be general. For Q3 and Q4, I think the discussion on how to choose k and d is not sufficient in the paper.

The reviews were mixed, but given the competitive nature of the conference, this paper probably doesn't make the threshold. Since the paper deals with adaptive dimensionality reduction, the following paper seems quite relevant: Lee-Ad Gottlieb, Aryeh Kontorovich, Robert Krauthgamer: Adaptive metric dimensionality reduction.

The robust principal component analysis (RPCA) problem seeks to separate low-rank trends from sparse outlierswithin a data matrix, that is, to approximate a n\times d matrix D as the sum of a low-rank matrix L and a sparse matrix S .We examine the robust principal component analysis (RPCA) problem under data compression, wherethe data Y is approximately given by (L S)\cdot C, that is, a low-rank sparse data matrix that has been compressed to size n\times m (with m substantially smaller than the original dimension d) via multiplication witha compression matrix C . We give a convex program for recovering the sparse component S along with the compressed low-rank component L\cdot C, along with upper bounds on the error of this reconstructionthat scales naturally with the compression dimension m and coincides with existing results for the uncompressedsetting m d . Our results can also handle error introduced through additive noise or through missing data.The scaling of dimension, compression, and signal complexity in our theoretical results is verified empirically through simulations, and we also apply our method to a data set measuring chlorine concentration acrossa network of sensors, to test its performance in practice.

This paper addresses the problem of estimating the positions of points from distance measurements corrupted by sparse outliers. Specifically, we consider a setting with two types of nodes: anchor nodes, for which exact distances to each other are known, and target nodes, for which complete but corrupted distance measurements to the anchors are available. To tackle this problem, we propose a novel algorithm powered by Nystr\"om method and robust principal component analysis. Our method is computationally efficient as it processes only a localized subset of the distance matrix and does not require distance measurements between target nodes. Empirical evaluations on synthetic datasets, designed to mimic sensor localization, and on molecular experiments, demonstrate that our algorithm achieves accurate recovery with a modest number of anchors, even in the presence of high levels of sparse outliers.

Suppose certain data points are overly contaminated, then the existing principal component analysis (PCA) methods are frequently incapable of filtering out and eliminating the excessively polluted ones, which potentially lead to the functional degeneration of the corresponding models. To tackle the issue, we propose a general framework namely robust weight learning with adaptive neighbors (RWL-AN), via which adaptive weight vector is automatically obtained with both robustness and sparse neighbors. More significantly, the degree of the sparsity is steerable such that only exact k well-fitting samples with least reconstruction errors are activated during the optimization, while the residual samples, i.e., the extreme noised ones are eliminated for the global robustness. Additionally, the framework is further applied to PCA problem to demonstrate the superiority and effectiveness of the proposed RWL-AN model.

In the current data-intensive era, big data has become a significant asset for Artificial Intelligence (AI), serving as a foundation for developing data-driven models and providing insight into various unknown fields. This study navigates through the challenges of data uncertainties, storage limitations, and predictive data-driven modeling using big data. We utilize Robust Principal Component Analysis (RPCA) for effective noise reduction and outlier elimination, and Optimal Sensor Placement (OSP) for efficient data compression and storage. The proposed OSP technique enables data compression without substantial information loss while simultaneously reducing storage needs. While RPCA offers an enhanced alternative to traditional Principal Component Analysis (PCA) for high-dimensional data management, the scope of this work extends its utilization, focusing on robust, data-driven modeling applicable to huge data sets in real-time. For that purpose, Long Short-Term Memory (LSTM) networks, a type of recurrent neural network, are applied to model and predict data based on a low-dimensional subset obtained from OSP, leading to a crucial acceleration of the training phase. LSTMs are feasible for capturing long-term dependencies in time series data, making them particularly suited for predicting the future states of physical systems on historical data. All the presented algorithms are not only theorized but also simulated and validated using real thermal imaging data mapping a ship's engine.