Core Concepts

New efficient algorithms for compressing biased input sequences into accurate summaries of a target distribution, achieving better-than-i.i.d. error rates.

Abstract

The content introduces a suite of new procedures for compressing a biased input sequence into an accurate summary of a target distribution P.
For equal-weighted compression, the key contributions are:
Stein Kernel Thinning (SKT): Combines the greedy bias correction of Stein thinning with the unbiased compression of kernel thinning to produce √n summary points with e^O(n^-1/2) MMD error in O(n^2) time. This is a significant improvement over the Ω(n^-1/2) error of standard i.i.d. sampling.
Low-rank SKT (LSKT): Achieves the same e^O(n^-1/2) MMD guarantee as SKT in sub-quadratic o(n^2) time by combining a scalable summarization technique with a new low-rank debiasing procedure.
For simplex-weighted and constant-preserving compression, the key contributions are:
Stein Recombination (SR) and Low-rank SR (LSR): Match the guarantees of SKT using as few as poly-log(n) weighted points by combining Stein thinning, low-rank debiasing, and a new recombination technique.
Stein Cholesky (SC) and Low-rank SC (LSC): Also match the SKT guarantees using poly-log(n) constant-preserving weighted points by combining Stein thinning, low-rank debiasing, and a new Cholesky-based compression scheme.
The algorithms are shown to effectively correct for biases due to burn-in, approximate MCMC, and tempering, outperforming baseline methods.

Stats

The input points Sn = (xi)n_i=1 are assumed to be the iterates of a homogeneous ϕ-irreducible geometrically ergodic Markov chain targeting a distribution Q, with tails no lighter than the target distribution P.
The kernel kP satisfies Assumption 1 (mean-zero kernel) and the (α, β)-kernel condition, which captures the smoothness of the kernel.
The input point radii Rn = max_i ∥x_i∥_2 ∨ 1 and kernel norms ∥kP∥_n = max_i kP(x_i, x_i) are assumed to be slow-growing, i.e., Rn = O((log n)^γ) and ∥kP∥_n = e^O(1) for some γ ≥ 0.

Quotes

"Remarkably, modern compression methods can summarize a distribution more succinctly than i.i.d. sampling."
"Much more commonly, one only has access to n biased sample points approximating a wrong distribution Q. Such biases are a common occurrence in Markov chain Monte Carlo (MCMC)-based inference due to tempering, burn-in, or approximate MCMC."
"Underlying these advances are new guarantees for the quality of simplex-weighted coresets, the spectral decay of kernel matrices, and the covering numbers of Stein kernel Hilbert spaces that may be of independent interest."

Key Insights Distilled From

by Lingxiao Li,... at **arxiv.org** 04-19-2024

Deeper Inquiries

The proposed debiased compression methods can be extended to handle non-Euclidean data domains, such as graphs or manifolds, by leveraging techniques from kernel methods and spectral graph theory.
For graphs, one approach is to define a kernel function that captures the similarity between nodes based on their graph structure. This kernel can be used in the debiased compression algorithms in a similar way to how it is used for Euclidean data. The WeightedRPCholesky algorithm, for example, can be adapted to work with graph data by constructing a kernel matrix that reflects the graph's connectivity.
For manifolds, techniques from manifold learning and kernel methods can be employed. By defining a kernel function that operates on the manifold's intrinsic geometry, the debiased compression methods can be applied to summarize the distribution on the manifold. Low-rank debiasing can be extended to handle the curvature and local structure of the manifold, allowing for efficient compression while preserving the underlying geometry.
In both cases, the key is to define an appropriate kernel function that captures the relevant relationships in the data domain and to adapt the debiased compression algorithms to operate in the non-Euclidean space.

The theoretical limits of debiased compression are influenced by the complexity of the target distribution and the bias in the input sequence. Tighter error bounds can potentially be achieved under additional assumptions on the target distribution or input sequence.
One way to improve error bounds is by incorporating more information about the target distribution into the compression algorithms. For example, if the target distribution has specific structural properties or symmetries, these can be exploited to design more efficient debiased compression methods. Additionally, refining the assumptions about the input sequence, such as the rate of convergence of the Markov chain or the level of bias, can lead to tighter error bounds.
Furthermore, exploring the trade-offs between computational complexity and error bounds can help identify the optimal balance for debiased compression. By analyzing the fundamental limits of the problem and considering the specific characteristics of the data and distribution, it may be possible to achieve even tighter error bounds under certain conditions.

The low-rank debiasing technique can be further improved or generalized to other settings beyond distribution compression by exploring the following avenues:
Adaptive Rank Selection: Developing algorithms that can adaptively select the rank parameter based on the characteristics of the input sequence and the target distribution. This adaptive approach can improve the efficiency and accuracy of the debiasing process.
Incorporating Domain Knowledge: Integrating domain-specific knowledge or constraints into the debiasing algorithms to enhance their performance. For example, incorporating constraints on the weights or the structure of the input data can lead to more effective debiased compression.
Extension to Dynamic Data: Generalizing the low-rank debiasing technique to handle dynamic or streaming data settings where the input sequence evolves over time. This extension would require updating the debiasing process in real-time to adapt to changes in the data distribution.
Application to Transfer Learning: Exploring the application of low-rank debiasing in transfer learning scenarios where knowledge from a source domain is leveraged to improve learning in a target domain. This extension can enhance the generalization capabilities of the debiasing technique.
By further refining the low-rank debiasing technique and exploring these extensions, it can be applied to a wider range of scenarios and contribute to more efficient and accurate data compression and analysis tasks.

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