[Paper Review] Quasi-metric spaces with measure

Aleksandar Stojmirović|ArXiv.org|Nov 6, 2003

Fixed Point Theorems AnalysisMathematics19 references22 citations

TL;DR

This paper extends concentration of measure phenomena from metric measure spaces (mm-spaces) to quasi-metric spaces with probability measures (pq-spaces), demonstrating that high-dimensional pq-spaces behave nearly like mm-spaces. It shows that common biological sequence similarity measures can be converted into quasi-metrics, and establishes exponential concentration bounds analogous to those in classical Lévy families, using asymmetric Hamming cubes and penalty-based methods.

ABSTRACT

The phenomenon of concentration of measure on high dimensional structures is usually stated in terms of a metric space with a Borel measure, also called an mm-space. We extend some of the mm-space concepts to the setting of a quasi-metric space with probability measure (pq-space). Our motivation comes from biological sequence comparison: we show that many common similarity measures on biological sequences can be converted to quasi-metrics. We show that a high dimensional pq-space is very close to being an mm-space.

Motivation & Objective

• To generalize concentration of measure concepts from mm-spaces to quasi-metric spaces with probability measures (pq-spaces).
• To demonstrate that high-dimensional pq-spaces are structurally close to mm-spaces in terms of concentration behavior.
• To show that common biological sequence similarity measures (e.g., Smith-Waterman) can be reformulated as quasi-metrics.
• To derive exponential concentration bounds for pq-spaces using asymmetric metrics and penalty functions.
• To establish that pq-spaces inherit key concentration properties, such as Lévy family behavior, under natural conditions.

Proposed method

• Define a quasi-metric space with a probability measure (pq-space), generalizing the mm-space framework.
• Introduce left, right, and associated metric balls and neighborhoods using the quasi-metric and its conjugate.
• Construct an asymmetric Hamming cube by defining a non-symmetric base quasi-metric on a binary alphabet.
• Apply Talagrand's penalty method to derive concentration bounds for pq-spaces using the ℓ₁-type quasi-metric on product spaces.
• Use the associated metric to relate pq-space concentration to classical mm-space results.
• Prove that the concentration function of a pq-space satisfies exponential bounds similar to those in Lévy families.

Experimental results

Research questions

• RQ1 Can concentration of measure be meaningfully extended from metric measure spaces to quasi-metric spaces with measure?
• RQ2 How do biological sequence similarity measures, such as Smith-Waterman scores, relate to quasi-metric structures?
• RQ3 To what extent do high-dimensional pq-spaces resemble mm-spaces in terms of concentration behavior?
• RQ4 What exponential bounds can be derived for left and right concentration functions in pq-spaces?
• RQ5 Can penalty-based methods from Talagrand’s work be adapted to derive concentration inequalities in pq-spaces?

Key findings

• The sequence of asymmetric Hamming cubes forms a normal Lévy family, with concentration function bounded by $ \exp(-2\varepsilon^2 n) $, matching the classical Hamming cube bound.
• The concentration function for the pq-space $ (\Sigma^n, q_n, \mu_n) $ satisfies $ \alpha_{(\Sigma^n, q_n, \mu_n)}(\varepsilon) \leq 2\exp\left(-\frac{n\varepsilon^2}{2}\right) $, showing strong concentration.
• For a product pq-space with quasi-metric $ q(x,y) = \sum_{i\leq N} q_\Omega(x_i, y_i) $, the concentration function is bounded by $ \alpha(\varepsilon) \leq 2\exp\left(-\min\left(\frac{\varepsilon^2}{8N\|q_\Omega\|_2^2}, \frac{\varepsilon}{2\|q_\Omega\|_\infty}\right)\right) $.
• The left and right concentration functions in pq-spaces inherit symmetric bounds due to the symmetry of the norms involved.
• The associated metric $ \hat{q} $ of a pq-space ensures that the pq-space is topologically close to a metric space, enabling transfer of concentration results.
• Biological sequence similarity measures, including Smith-Waterman scores, can be transformed into generalized weighted quasi-metrics, enabling their use in pq-space frameworks.