Random-k Approximation Methods

• Random-k Approximation is a collection of randomized algorithms and geometric methods that construct approximate representations of complex mathematical objects using only k samples.
• It provides rigorous error analyses and probabilistic guarantees in high-dimensional geometry, optimization, and statistical estimation.
• The approach underpins efficient spectral algorithms and randomized rounding schemes, achieving polynomial complexity and near-optimal error bounds.

Random-k approximation encompasses a collection of randomized algorithms, geometric methods, and analytic bounds for constructing approximate representations of complex mathematical objects—functions, convex bodies, distributions, eigenspaces, and more—using only $k$ samples, parameters, or signals. These approaches are characterized by rigorous error analysis, probabilistic guarantees, and broad applicability in high-dimensional geometry, optimization, statistical estimation, and computational mathematics. The term “Random-%%%%1%%%% Approximation” thus refers to both structural theorems about sampling (e.g., of convex bodies or graphs), as well as algorithmic schemes for optimal approximation under cardinality constraints.

1. Random-k Approximation in Convex Geometry

In geometric analysis, random-k approximation typically addresses the minimal sample size necessary for a random subset $X = \{x_1, \dots, x_k\}$ of a convex body $K \subset \mathbb{R}^n$ to yield, with high probability, a scaled convex hull that covers $K$. The main theorem of Brazitikos–Chasapis–Hioni (Brazitikos et al., 2015) establishes that for a convex body $K$ of volume one and center of mass at the origin, there exist absolute constants $\alpha, c_1 > 0$ such that sampling $k = \lceil \alpha n \rceil$ independent points uniformly from $K$ yields

$$\mathbf{P}\left[\,K \subset c_1\, n\, \text{conv}(X)\,\right] \ge 1 - e^{-n}$$

This result is derived by reducing $K$ to isotropic position, analyzing “one-sided” $L_q$ centroid bodies $Z_q^+(K)$, and bounding the measure of all facets of the resulting polytope not intersecting large centroid bodies via the Paley–Zygmund inequality and union bounds.

The scaling factors and constants emerge from volumetric relationships between $K$, its $L_q$ centroid bodies, and Euclidean balls, ultimately yielding a universal quadratic upper bound for the vertex index of $K$: $\mathrm{vi}(K) \le C n^2$ where $\mathrm{vi}(K)$ is the minimal weighted Minkowski sum of scaling factors necessary to represent $K$ as a convex hull.

2. Hausdorff Approximations via Random Sampling

Recent advances in the probabilistic approximation of convex bodies also analyze the convergence—both in expectation and distribution—of random polytopes to their parent body in the Hausdorff metric. For smooth convex bodies $K \subset \mathbb{R}^d$ with strictly positive Gaussian curvature, sampling $n$ points from the optimal density $h_\kappa(x) \propto \sqrt{\kappa_K(x)}$ and forming $K_n = \text{conv}\{X_1, \dots, X_n\}$ yields: $$E_n = \delta_H(K_n, K) \sim C \left( \frac{\log n}{n} \right)^{2/(d-1)}$$ with fluctuations governed by an explicit Gumbel distribution (Sonnleitner, 22 Aug 2025). In two dimensions, for polygons, sharp constants are extracted in terms of perimeter and interior angles, revealing asymptotic error scaling as $(\log M)/M$ for regular $M$-gons (Prochno et al., 2024). Both boundary and interior sampling yield distinct regimes, with random sampling achieving the same leading order as optimal deterministic configurations.

3. Random-k Approximation in Functional and Hilbert Spaces

In functional analysis, random-k approximation refers to the construction of randomized sketches for approximating a scattered data interpolation problem in infinite-dimensional Hilbert spaces. Let $A$ be a set, $V$ a Hilbert space, and $H$ a Hilbert space of functions $f:A\to V$. For dataset $\{(x_i, y_i)\}_{i=1}^n \subset A \times V$, consider the functional: $$u(f) = \frac{q}{2}\Vert f\Vert_{H}^2 + \frac{1-q}{2n}\sum_{i=1}^{n}\Vert f(x_i)-y_i\Vert_{V}^2$$ where $0 < q < 1$. The randomized approximation constructs, for each $k \ge 2$, a random function

$$F_k = \sum_{h=1}^{N_k} \Lambda_{k,h}\, \Phi(x_{I_h}, \mathcal{E}_h)$$

using Riesz representatives $\Phi(x,v)\in H$ and random coefficients, with $N_k$ binomially distributed and $I_h$ uniformly chosen indices. The expected $H$-norm error decays as $O(1/k)$, providing Monte Carlo-type guarantees even in absence of metric or measurability structure on $A$ (Yeressian, 2019). This construction is based on stochastic gradient descent in Hilbert space and applies to infinite-dimensional $V$.

4. Random-k Algorithms in Spectral and Statistical Approximation

Random-k approximation is central to efficient spectral algorithms. For any symmetric graph Laplacian $L \in \mathbb{R}^{n\times n}$, the span of the first $k$ eigenvectors $V_k$ can be exactly recovered, with probability one, by applying the orthogonal projector $P_k$ to $k$ independent Gaussian random signals $R$: $M = P_k R = V_k V_k^T R$ Rank and subspace are preserved, and numerical stability is ensured by concentration of singular values in Gaussian matrices (Paratte et al., 2016). In practice, polynomial filters (Jackson–Chebyshev) approximate $P_k$, yielding $O(n k^2)$ algorithms for massive graphs.

In randomized approximation of statistical properties of randomly weighted graphs, the expected value and higher moments of distance-cumulative properties (minimum spanning tree, diameter, etc.) are fully polynomial-time randomized approximation scheme (FPRAS) computable for fixed $k$. The key scheme decomposes moments into weighted sums of tail probabilities $P[Y>x]$ over geometric grids, each estimated via network reliability and DNF satisfiability algorithms, evading the curse of variance that plagues naive Monte Carlo (0908.0968).

5. Randomized Rounding and Factor-Revealing LPs

Random-k approximation is instrumental in the design of randomized rounding schemes for integer programming, notably for $k$-level uncapacitated facility location with penalties (UFLWP). Each possible scaling parameter $\gamma > 1$ defines a rounding algorithm $A(\gamma)$; by constructing a small factor-revealing LP, one extracts a probability distribution over these $\gamma$ (and over alternative algorithms such as JMS) so as to minimize worst-case approximation factor (Byrka et al., 2013). The randomized algorithm achieves expected cost

$$E[A] \le \alpha_k\,\text{(optimal fractional cost)}$$

with $\alpha_k$ obtained as the optimal LP value. This removes the need for ad-hoc density construction and generalizes prior approaches to the multi-level, penalized setting.

6. Algorithmic Frameworks, Complexity, and Empirical Performance

Random-k approximation algorithms universally exhibit polynomial sample-size or runtime guarantees for fixed $k$ (dimension or signal count). In Kolmogorov approximation of discrete distributions, state-of-the-art optimal algorithms achieve $O(n\log n)$ complexity and $O(1/m)$ error scaling, maintaining minimal support cardinality (Cohen et al., 2022). In FPRAS for random graph properties, the complexity is polynomial in $n$, $1/\epsilon$, and $\log(1/\delta)$ for any fixed $k$ (0908.0968).

Empirical observations confirm that, across domains, random-k algorithms perform close to theoretical bounds, with error rates asymptotically matching deterministic optimal schemes, sharp concentration inequalities (sub-Gaussian tails), and explicit high-probability guarantees.

7. Open Questions and Extensions

Despite its successes, random-k approximation leaves open multiple avenues:

• Determination of sharp constants and limit laws for Hausdorff approximation in higher dimensions, especially for non-smooth $K$ (Prochno et al., 2024, Sonnleitner, 22 Aug 2025).
• Adaptive or leverage-score based spectral filtering for dynamic or streaming graphs (Paratte et al., 2016).
• Generalization to non-symmetric data matrices, richer metric spaces, or “online” constructions in scheduling and probabilistic network analysis.
• Comparisons between different metrics of approximation (symmetric difference vs Hausdorff) and their respective optimality regimes.
• Limitations where the parameter $k$ itself is unbounded or scales with error tolerance, potentially violating polynomial complexity (cf. tail-approximation grid sizes in FPRAS (0908.0968)).

In summary, random-k approximation delivers a unified analytic and algorithmic approach for efficient, probabilistic, and optimally bounded approximation schemes across diverse mathematical and computational fields. It leverages probabilistic tools (Paley–Zygmund, union bounds, DNF-FPRAS) and convexity structures (isotropic position, centroid bodies, Riesz representatives), providing robust guarantees and practical algorithms for high-dimensional and randomized settings.