Convex Cone Sparsification Function

• The sparsification function is a measure that determines the minimum number of elements required to approximate sums in a convex cone within a specified relative error.
• It generalizes spectral sparsification from positive semidefinite matrices to arbitrary convex cones using tools from convex analysis and interior-point theory.
• Barrier-based methods establish explicit upper bounds on the sparsifier size, enabling efficient sparse approximations in large-scale conic optimization problems.

The sparsification function of a convex cone quantifies the minimal support size required to approximate arbitrary sums of elements from the cone to within a prescribed order-relative error. This concept generalizes spectral sparsification from sums of positive semidefinite matrices to sums within arbitrary convex cones, using tools from convex analysis and interior-point theory. The sparsification function provides worst-case bounds that are intrinsic to the geometric and barrier properties of the cone.

1. Foundational Definitions

Let $K \subseteq \mathbb{R}^n$ be a closed convex cone with the cone-induced partial order

$x\;_K\!y \iff y - x\;\in K.$

The relative interior is denoted $\RelInt(K)$.

$\varepsilon$-Sparsifier: Given $x_1,\ldots,x_m \in K$ with $e = \sum_{i=1}^m x_i \in \RelInt(K)$ and $0 < \varepsilon < 1$, an $\varepsilon$-sparsifier of $e$ comprises a subset $S \subseteq \{1, \ldots, m\}$ and weights $\{\lambda_i > 0 : i \in S\}$ so that

$(1-\varepsilon)\,e\;_K\; \sum_{i \in S}\lambda_i x_i\;_K\; (1+\varepsilon)\,e.$

Sparsification Function: A function $\alpha : (0,1) \to \mathbb{R}_+$ is a sparsification function for $K$ if, for every collection $\{x_i\} \subset K$ summing to $e \in \RelInt(K)$ and every $\varepsilon \in (0,1)$, there exists an $\varepsilon$-sparsifier $S, \{\lambda_i\}$ with $|S| \le \alpha(\varepsilon)$. The sparsification function is defined as

$$sp_K(\varepsilon) := \inf_{\alpha \in \mathcal{F}_K} \alpha(\varepsilon),$$

where $\mathcal{F}_K$ is the set of feasible such $\alpha$. Carathéodory’s theorem for cones implies $sp_K(\varepsilon) \le \dim(K)$ (Saunderson, 26 Dec 2025).

2. Upper Bounds: Barrier-Based Results

For a proper cone $K$ (closed, pointed, full-dimensional) that admits a $\nu$-logarithmically homogeneous self-concordant barrier, i.e., a $C^3$ convex $F:\Int(K)\to \mathbb{R}$ satisfying

$$F(tx) = F(x)-\nu \ln t, \quad |D^3 F(x)[u,u,u]| \le 2 [ D^2 F(x)[u,u] ]^{3/2},$$

the following bounds are established:

• General Case: If $K$ admits such a barrier,

$$sp_K(\varepsilon) \le \Big\lceil\, (4\nu/\varepsilon)^2 \Big\rceil, \quad \forall \varepsilon \in (0,1).$$

• Pairwise Self-Concordant Case: If, additionally,

$0 \le -D^3F(x)[v,u,u] \le 2 D^2F(x)[v,u]|u|_x,$

for all $x \in \Int(K), u,v \in K$, where $|u|_x$ is the minimal $t$ with $-tx\,_K u\,_K tx$, then

$$sp_K(\varepsilon) \le \Big\lceil \, 4\nu / \varepsilon^2 \Big\rceil, \quad \forall \varepsilon \in (0,1).$$

Every hyperbolicity cone, and in particular the positive semidefinite cone $S^d_+$, satisfies the pairwise condition with $\nu = d$, recovering the Batson–Spielman–Srivastava bound $O(d/\varepsilon^2)$ (Saunderson, 26 Dec 2025).

3. Algorithmic Proof Sketches via Barrier Methods

3.1 Frank–Wolfe Construction

For the $\lceil(4\nu/\varepsilon)^2\rceil$ upper bound, classical self-concordance yields weights $\{\mu_i > 0\}$ such that

$$e = \sum_{i=1}^m \mu_i x_i, \quad -\sum_i D F(e)[x_i] = \nu,$$

and thus $w_i = -D F(e)[x_i] / \nu$ with $\sum_i w_i=1$. Set $\tilde{x}_i = x_i / w_i$. Defining the convex set $\mathcal{X} = \operatorname{Conv}\{\tilde{x}_i\}$ and the quadratic objective $f(z) = \tfrac{1}{2}\|z-e\|_e^2$ with $\|u\|_e^2 = D^2 F(e)[u,u]$, the Frank–Wolfe algorithm produces, after $T \approx 8\nu^2/\varepsilon^2$ iterations, a point $z_T \in \mathcal{X}$ with $\|z_T - e\|_e \le \varepsilon$. Self-concordance properties ensure this yields an $\varepsilon$-sparsifier of size $T$.

3.2 BSS-style Iteration for Pairwise Barriers

For the sharper $\lceil 4\nu/\varepsilon^2 \rceil$ bound, define "upper" and "lower" barrier potentials

$$\Phi^{u,e}(x) = -D F(ue - x)[e], \quad \Phi_{\ell,e}(x) = -D F(x - \ell e)[e].$$

Following a Batson–Spielman–Srivastava-type greedy iteration and using the pairwise self-concordance condition, at each step a new sparsifier component is added without increasing the barrier potentials. After $T$ steps, the resulting normalized sum is an $\varepsilon$-sparsifier with $T$ terms (Saunderson, 26 Dec 2025).

4. Concrete Instance: Positive Semidefinite Cone

For $K = S^d_+$, the standard logarithmic barrier is $F(X) = -\ln\det X$ with parameter $\nu = d$. The pairwise self-concordance condition holds (e.g., by Loewner’s theorem). Hence,

$$sp_{S^d_+}(\varepsilon) \le \left\lceil 4d / \varepsilon^2 \right\rceil,$$

exactly matching the dimension-dependent sparsification originally established for matrix-valued spectral sparsification.

5. Geometric Operations and Monotonicity

If a convex set $C$ has a proper $K$-lift, meaning $C = \pi(K \cap L)$ for some linear space $L$ meeting $\RelInt(K)$ and linear map $\pi$, then

$sp_C(\varepsilon) \le sp_K(\varepsilon).$

In particular, intersection with a hyperplane meeting $\RelInt(K)$, linear projections, convex lifts, and extended formulations all do not increase the sparsification function. This demonstrates stability under standard convex-geometric operations and suggests intrinsic geometric monotonicity of the sparsification function (Saunderson, 26 Dec 2025).

6. Applications to Conic Optimization

For covering-type conic programs

$\min_{y \ge 0} \langle b, y \rangle \quad \text{s.t.} \quad \sum_{i=1}^m y_i a_i\;_K\;c,$

where $c = \sum_i c_i$, the $m$ constraints $\{a_i\}$ can be replaced by an $\varepsilon$-sparsifier of size $sp_K(\varepsilon)$, yielding a near-optimal sparse solution $\tilde{y}$ with $$|\operatorname{supp}(\tilde{y})| \le sp_K(\varepsilon)$$.

For packing-type duals,

$$\max_{x \in K^*} \langle c, x \rangle \quad \text{s.t.} \quad \langle a_i, x\rangle \le b_i,$$

replacing the cost vector $c$ by its sparsifier $c'$ alters the optimum by at most a $1 \pm \varepsilon$ factor. Thus, cone sparsification reduces the support size of near-optimal feasible points, with implications for the acceleration of first-order or combinatorial algorithms in large-scale conic optimization settings (Saunderson, 26 Dec 2025).