Current Trends in High-Dimensional Statistical Learning and Robust Algorithms

The recent advancements in high-dimensional statistical learning and robust algorithms have been notably focused on enhancing the efficiency and reliability of computational methods. A significant trend is the development of algorithms that leverage sum-of-squares (SoS) techniques to achieve near-optimal guarantees in various statistical tasks, such as robust mean estimation, clustering, and covariance estimation. These algorithms are particularly noteworthy for their ability to handle subgaussian distributions, which are prevalent in many real-world datasets.

Another emerging area is the exploration of lower bounds within the SoS framework, which provides insights into the inherent limitations of current algorithms. This research highlights the need for novel techniques to overcome these barriers, particularly in tasks like Non-Gaussian Component Analysis (NGCA) and density estimation. The introduction of new analytical methods, such as those based on polynomial degradation analysis, is also advancing the field by offering faster solutions to problems like average-case orthogonal vectors and closest pair problems.

In the realm of computational efficiency, there is a growing emphasis on reducing space complexity, especially in streaming settings, without compromising on the number of samples required. This is exemplified by recent work on testing identity of distributions under Kolmogorov distance, which demonstrates that polylogarithmic space can be achieved with optimal sample usage.

Noteworthy Papers

1. SoS Certifiability of Subgaussian Distributions and its Algorithmic Applications: Introduces a novel condition for subgaussian distributions that yields efficient learning algorithms for a wide range of high-dimensional tasks.
2. Sum-of-squares lower bounds for Non-Gaussian Component Analysis: Provides the first super-constant degree SoS lower bound for NGCA, highlighting a significant information-computation tradeoff.
3. Testing Identity of Distributions under Kolmogorov Distance in Polylogarithmic Space: Demonstrates a significant reduction in space complexity for distribution testing without compromising sample efficiency.
4. Faster Algorithms for Average-Case Orthogonal Vectors and Closest Pair Problems: Offers a new approach to solving these problems faster in the average case using polynomial methods.