pstabletype

pstabletype is a conceptual term in probability and data analysis used to describe a class of data types or random variables that exhibit stability under linear combinations, governed by a stability index p in the range (0, 2]. The idea draws on p-stable distributions, a family that includes familiar cases such as the Gaussian (p = 2) and the Cauchy (p = 1). pstabletype is used to characterize datasets whose projections preserve information about their original norms, making them suitable for certain forms of dimensionality reduction and norm estimation.

Definition and properties: A random vector X is said to be of pstabletype if, for any finite collection of independent copies X1, X2, ..., Xk with the same distribution as X, and any real coefficients a1, a2, ..., ak, the linear combination sum ai Xi has the same distribution as (sum |ai|^p)1/p times a standard p-stable variable. In practical terms, projections of data drawn from a pstabletype distribution onto random vectors with p-stable distributions preserve the L_p norm structure of the data up to a scale, enabling efficient estimation of norms and distances in high dimensions.

Applications: pstabletype concepts underpin randomized linear algebra methods that aim to approximate L_p norms and distances through low-dimensional projections. They are used in sketching and streaming algorithms, dimensionality reduction, and certain hashing schemes. The framework aligns with the use of p-stable distributions in projecting data so that the projected values reflect the original data’s L_p geometry, a technique highlighted in foundational work on p-stable random projections.

Limitations and notes: Distributions with stability index p may have heavy tails and may lack moments of order higher than p, which influences estimation and convergence properties. Estimating the index p and applying pstabletype-based methods require careful statistical handling and adequate sample sizes.

See also: p-stable distributions, L_p norms, randomized projections, stable distributions, sketching and hashing in data streams.