Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Revisiting RIP guarantees for sketching operators on mixture models

Ayoub Belhadji, Rémi Gribonval

TL;DR

This work revisits RIP guarantees for sketching operators in compressive mixture modeling, challenging prior requirements for importance sampling in random Fourier features. By deriving deterministic bounds on the RIP constant that depend only on the sketch’s frequencies, the authors obtain concentration results and RIP guarantees for both random and structured (e.g., block-i.i.d.) Fourier sketches, with sketch size scaling roughly as $m = \mathcal{O}(k^{2} d)$ under broad conditions. A key contribution is a dipole-based decomposition that reduces the analysis to finite-dimensional empirical processes, enabling relaxation of growth conditions on the weighting function $w$. The paper also establishes lower bounds on variance and demonstrates limitations of existing approaches, while highlighting open questions about achieving $\mathcal{O}(k d)$ scaling in practice. Together, these results deepen the theoretical understanding of sketching-based inference for mixture models and pave the way for deterministic and structured sketching approaches in high-dimensional settings.

Abstract

In the context of sketching for compressive mixture modeling, we revisit existing proofs of the Restricted Isometry Property of sketching operators with respect to certain mixtures models. After examining the shortcomings of existing guarantees, we propose an alternative analysis that circumvents the need to assume importance sampling when drawing random Fourier features to build random sketching operators. Our analysis is based on new deterministic bounds on the restricted isometry constant that depend solely on the set of frequencies used to define the sketching operator; then we leverage these bounds to establish concentration inequalities for random sketching operators that lead to the desired RIP guarantees. Our analysis also opens the door to theoretical guarantees for structured sketching with frequencies associated to fast random linear operators.

Revisiting RIP guarantees for sketching operators on mixture models

TL;DR

This work revisits RIP guarantees for sketching operators in compressive mixture modeling, challenging prior requirements for importance sampling in random Fourier features. By deriving deterministic bounds on the RIP constant that depend only on the sketch’s frequencies, the authors obtain concentration results and RIP guarantees for both random and structured (e.g., block-i.i.d.) Fourier sketches, with sketch size scaling roughly as under broad conditions. A key contribution is a dipole-based decomposition that reduces the analysis to finite-dimensional empirical processes, enabling relaxation of growth conditions on the weighting function . The paper also establishes lower bounds on variance and demonstrates limitations of existing approaches, while highlighting open questions about achieving scaling in practice. Together, these results deepen the theoretical understanding of sketching-based inference for mixture models and pave the way for deterministic and structured sketching approaches in high-dimensional settings.

Abstract

In the context of sketching for compressive mixture modeling, we revisit existing proofs of the Restricted Isometry Property of sketching operators with respect to certain mixtures models. After examining the shortcomings of existing guarantees, we propose an alternative analysis that circumvents the need to assume importance sampling when drawing random Fourier features to build random sketching operators. Our analysis is based on new deterministic bounds on the restricted isometry constant that depend solely on the set of frequencies used to define the sketching operator; then we leverage these bounds to establish concentration inequalities for random sketching operators that lead to the desired RIP guarantees. Our analysis also opens the door to theoretical guarantees for structured sketching with frequencies associated to fast random linear operators.
Paper Structure (72 sections, 30 theorems, 263 equations, 3 figures)

This paper contains 72 sections, 30 theorems, 263 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main tools
  3. Sketching operator
  4. Separated mixture model, normalized secant set, and dipoles
  5. Kernel coherence
  6. Location-based families and shift-invariant kernels
  7. Random Fourier features
  8. Existing results and their limitations
  9. Main results
  10. On the necessity of conditions \ref{['eq:phi_omega_conditions']}
  11. Sharp deterministic bounds on $\delta(\mathcal{S}_{k}|\mathcal{A})$
  12. From the normalized secant set to normalized monopoles and balanced dipoles
  13. Expression using the supremum of certain empirical processes
  14. Lipschitz property and covering numbers
  15. Results for random sketching
  16. ...and 57 more sections

Key Result

Lemma 2

Let $\pi, \pi' \in \mathfrak{G}_{k}$. There exist $\ell \leq 2k$ nonzero dipoles $(\iota_{l})_{l \in [\ell]}$ that are pairwise $1$-separated and satisfy

Figures (3)

  • Figure 1: (top) Histogram of $\psi(\omega)$ when $\omega \sim \mathcal{N}(0,s^{-2}\mathbb{I}_d)$; (bottom) Empirical graph of $\mathbb{P}(\psi(\omega) - \mathbb{E}\psi(\omega) \geq \epsilon)$ and two candidate analytic bounds for $s =3$ and $d = 5$ (left), $d=100$ (right).
  • Figure 2: The term $m \times \mathbb{V}\|Ax_{k}\|^2$ (left) compared to the term $m \times \mathbb{V}\|\mathcal{A}\nu_{k}\|^2$ (right).
  • Figure 3: An illustration of the lower bound of Proposition \ref{['prop:lower_bound_psi_mm']} (left) for three choices of $w$ (right): $w_0(\omega) = 1, \:\: w_{1}(\omega) = (1+\|\omega\|)^{-1}, \:\: w_{2}(\omega) = (\|\omega\|^4+1)(\|\omega\|^6+1)^{-1}$.

Theorems & Definitions (42)

  • Definition 1: Dipoles GrBlKeTr20
  • Lemma 2: GrBlKeTr20
  • Definition 3
  • Definition 4: Coherence GrBlKeTr20
  • Definition 5: Operator Coherence
  • Proposition 6: GrBlKeTr20
  • Example 1
  • Example 2
  • Definition 7
  • Definition 8
  • ...and 32 more