A Unified Theory of Sparsification

Abstract: We study the sparsifiability of \emph{real-valued codes}, a unifying abstraction that generalizes both combinatorial and continuous notions of sparsification, including spectral sparsification. In our setting, a code $C \subseteq \mathbb{R}<em>{\geq 0}<sup>m$ is simply a collection of nonnegative real-valued vectors, and for a parameter $ε&gt; 0$, a \emph{$(1 \pm ε)$-sparsifier} of $C$ is a subset $T \subseteq [m]$, together with weights $w \in \mathbb{R}</em>{\geq 0}<sup>T$, such that, for every $c \in C$, $$\sum_{i \in T} w_i c_i \in (1 \pm ε)\sum_{i=1}^m c_i$$. When $C \subseteq {0,1}<sup>m$, this specializes to code sparsification, and hence captures CSP sparsification, as studied by Khanna--Putterman--Sudan (SODA 2024, STOC 2025) and Brakensiek--Guruswami (STOC 2025). Similarly, for a graph $G=(V,E)$, if one defines $$C={c^{{(x)}:x\in\mathbb} R^{V}\subseteq\mathbb} R_{\geq 0}<sup>E$$ by $c<sup>{(x)}_{(u,v)}=(x_u-x_v)<sup>2$, then sparsifying $C$ is exactly spectral graph sparsification, as studied by Spielman--Teng (SICOMP 2011). Although the techniques driving combinatorial and continuous sparsification have traditionally been largely disjoint, our main result is a single structural theorem governing the sparsifiability of arbitrary real-valued codes $C\subseteq\mathbb{R}_{\geq 0}<sup>m$. The central parameter is \emph{continuous-valued non-redundancy} ($\mathrm{CVNRD}$), a real-valued analogue of non-redundancy that captures the largest approximately block-diagonal obstruction contained in $C$. Our theorem gives sparsifiers of size nearly-linear in $\mathrm{CVNRD}$, and shows that $\mathrm{CVNRD}$ is also a lower-bound obstruction for the broad class of coordinate-wise unbiased randomized sparsification schemes.

• The paper introduces continuous-valued non-redundancy (CVNRD), a block-diagonal obstruction that characterizes sparsifier size for real-valued codes up to polynomial factors in ε⁻¹ and polylogarithmic factors in the dimension.
• The framework unifies Boolean, spectral, hypergraph, submodular, and valued-CSP sparsification, including nearly tight Õ(n²/ε⁴) bounds for bounded submodular sums and near-linear bounds for higher-power graph and hypergraph energies.
• The paper replaces dependence on codeword counts with Sauer–Shelah and covering arguments, enabling results for infinite codes while leaving open whether the randomized lower bound extends to every sparsification method in the unbounded-range setting.

Overview

This paper develops a structural theory of sparsification for real-valued codes, unifying combinatorial sparsification (cut and CSP sparsifiers) with continuous sparsification (spectral graph and hypergraph sparsification) under a single abstraction. A code $C \subseteq \mathbb{R}_{\geq 0}^m$ is an arbitrary collection of nonnegative vectors; a $(1\pm\epsilon)$-sparsifier is a reweighted subset of coordinates preserving the weight of every codeword. The central contribution is a single parameter — continuous-valued non-redundancy ($\mathrm{CVNRD}$), measuring the largest approximately block-diagonal obstruction inside a code — that governs sparsifiability up to polynomial factors in $\epsilon^{-1}$ and polylogarithmic factors in $m$ (2607.16126).

The framework strictly generalizes prior work: Boolean codes $C \subseteq \{0,1\}^m$ recover the non-redundancy characterization of Brakensiek–Guruswami (STOC 2025); the spectral code $c^{(x)}_{(u,v)} = (x_u - x_v)^2$ recovers Spielman–Teng spectral sparsification; and the $t$-spectral code $c^{(x)}_{(u,v)} = |x_u-x_v|^t$ yields new results for higher powers of graph energies.

The main classification theorem

The paper defines three notions of sparsifiability: unweighted sparsifiability $US(C,\epsilon)$ (worst-case over coordinate restrictions), weighted sparsifiability $(1\pm\epsilon)$0, and random sparsifiability $(1\pm\epsilon)$1, where a random sparsifier is any distribution over weight functions satisfying only coordinate-wise unbiasedness $(1\pm\epsilon)$2. This model is deliberately broad: it permits arbitrary nonuniform, correlated sampling-and-reweighting schemes (the standard paradigm underlying most sparsification algorithms), excluding only deterministic constructions such as Batson–Spielman–Srivastava.

The main theorem states that, setting $(1\pm\epsilon)$3,

$(1\pm\epsilon)$4

and conversely,

$(1\pm\epsilon)$5

A $(1\pm\epsilon)$6 witness consists of disjoint coordinate blocks $(1\pm\epsilon)$7 with subcodes $(1\pm\epsilon)$8 such that each block is "completely shattered": for every subset $(1\pm\epsilon)$9, some codeword realizes a low value $\mathrm{CVNRD}$0 on $\mathrm{CVNRD}$1 and a high value $\mathrm{CVNRD}$2 on $\mathrm{CVNRD}$3, while off-block entries are small both cumulatively and entrywise. This is the real-valued analogue of a diagonal submatrix: for Boolean codes, every such witness forces exact block diagonality, so $\mathrm{CVNRD}$4 reduces to classical non-redundancy, recovering the qualitative content of Brakensiek–Guruswami's theorem as a corollary.

Two features distinguish this theorem from prior work. First, it pays no factor depending on $\mathrm{CVNRD}$5 — essential because real-valued codes may be infinite (as in spectral sparsification), and even finite obstructions can require exponentially many codewords. The paper demonstrates this with the code $\mathrm{CVNRD}$6: for $\mathrm{CVNRD}$7, every sparsifier needs $\mathrm{CVNRD}$8 coordinates, yet any finite witness requires $\mathrm{CVNRD}$9 codewords, since replacing nonzero entries by $\epsilon^{-1}$0 collapses the entire code to a single word. Second, the lower bound holds against all unbiased randomized schemes, not merely independent sampling.

Limitations of the general characterization

The lower bound in full generality is proved only for the randomized model rather than for all sparsifiers; whether $\epsilon^{-1}$1 always holds is left open, though the authors conjecture the gap is an artifact of the proof. This gap disappears in bounded-aspect-ratio settings, where lower bounds apply to arbitrary sparsifiers.

Bounded-aspect-ratio codes and applications

For codes $\epsilon^{-1}$2 with constant $\epsilon^{-1}$3, the paper introduces $\epsilon^{-1}$4, a cleaner obstruction requiring exact zeros off-block, and proves nearly tight bounds:

$\epsilon^{-1}$5

Two consequences follow. For submodular sparsification, sums of $\epsilon^{-1}$6-bounded submodular functions $\epsilon^{-1}$7 admit $\epsilon^{-1}$8-sparsifiers of size $\epsilon^{-1}$9, essentially settling the complexity for bounded functions: the best previous upper bound was $m$0 (Kenneth–Krauthgamer), while the known $m$1 lower bound via directed cuts shows $m$2 is likely optimal. Extension to arbitrary nonnegative submodular functions remains open.

For valued CSPs, the paper defines $m$3 — a predicate-level analogue of non-redundancy with complete blocks over pairs of values in $m$4 — and proves that worst-case sparsifier size satisfies $m$5 for constant $m$6. Notably, complete blocks arise even for simple predicates: the predicate $m$7 has $m$8, whereas its Boolean support projection has non-redundancy only $m$9. This shows that magnitude structure, not merely zero/nonzero patterns, fundamentally governs VCSP sparsifiability.

Spectral consequences

Applying the general theorem to graphs, the paper proves that the $C \subseteq \{0,1\}^m$0 of the spectral code of any $C \subseteq \{0,1\}^m$1-vertex graph is $C \subseteq \{0,1\}^m$2, via an elementary combinatorial argument: complete shattering forbids short cycles within each block (an odd cycle contradicts the telescoping sum $C \subseteq \{0,1\}^m$3, and even cycles are handled by designating one edge at the low level), while girth–density tradeoffs force short cycles across blocks once the witness exceeds $C \subseteq \{0,1\}^m$4 edges, violating the entrywise off-block condition. This yields:

• Higher-power graph spectra: for all $C \subseteq \{0,1\}^m$5, sparsifiers of size $C \subseteq \{0,1\}^m$6 preserving $C \subseteq \{0,1\}^m$7, improving on the prior best $C \subseteq \{0,1\}^m$8 of Jambulapati–Lee–Liu–Sidford.
• Hypergraph $C \subseteq \{0,1\}^m$9-spectral sparsification: near-linear-size sparsifiers for $c^{(x)}_{(u,v)} = (x_u - x_v)^2$0 with no assumptions on hyperedge sizes; for $c^{(x)}_{(u,v)} = (x_u - x_v)^2$1 this gives the first chaining-free proof of spectral hypergraph sparsification, and for $c^{(x)}_{(u,v)} = (x_u - x_v)^2$2 the first near-linear-size sparsifiers of this kind.

The hypergraph reduction uses the graph-theoretic Sauer–Shelah lemma of Cesa-Bianchi and Haussler to fix consistent "representative pairs" witnessing high energy, converting hyperedges into graph edges at a cost of $c^{(x)}_{(u,v)} = (x_u - x_v)^2$3.

Technique: Sauer-Shelah as the common core

Methodologically, the paper replaces the toolkit of Gilmer's entropy method, matrix Chernoff concentration, and Talagrand chaining with the Sauer–Shelah lemma and its relatives. The Boolean argument proceeds by iteratively peeling dense subcodes: if removing a set $c^{(x)}_{(u,v)} = (x_u - x_v)^2$4 is necessary to shrink the code below $c^{(x)}_{(u,v)} = (x_u - x_v)^2$5 codewords, then repeatedly invoking Sauer–Shelah extracts disjoint complete blocks; subsampling reduces off-block weight; pigeonhole arguments align off-diagonal patterns; greedy processing makes the collection block upper-triangular and then truly block-diagonal; and a final Sauer–Shelah invocation converts each surviving block into a complete subcode, yielding an identity matrix of size $c^{(x)}_{(u,v)} = (x_u - x_v)^2$6 and hence $c^{(x)}_{(u,v)} = (x_u - x_v)^2$7. The continuous analogue replaces codeword counts with sizes of $c^{(x)}_{(u,v)} = (x_u - x_v)^2$8-covers, using the Alon–Ben-David–Cesa-Bianchi–Haussler bound relating cover size to fat-shattering dimension. For unbounded aspect ratios, the proof analyzes auxiliary codes restricted to narrow value ranges $c^{(x)}_{(u,v)} = (x_u - x_v)^2$9, since cover-size bounds and Chernoff concentration both degrade with the ratio $t$0 — a dependence strong enough that when $t$1, even single-codeword preservation fails under naive sampling.

Conclusion

The paper establishes that a single block-diagonal obstruction parameter controls sparsification across Boolean, discrete, bounded-aspect-ratio, and fully general real-valued codes, with matching upper bounds and randomized-scheme lower bounds, and derives improved or first-of-their-kind sparsifiers for higher-order spectral energies, hypergraphs, bounded submodular sums, and valued CSPs. The principal open questions left by the work are whether $t$2 lower-bounds all sparsifiers (not just unbiased randomized ones) in the unbounded-aspect-ratio regime, and whether the submodular bound extends beyond bounded-range functions to arbitrary nonnegative submodular functions.