Sparse Direct-Product Testers

• The paper's main contribution is the development of sparse direct-product testers that efficiently verify if a function is nearly a direct-product encoding by using a constant number of queries.
• They employ explicit constructions of constant-degree simplicial complexes and hypergraphs, leveraging coboundary expansion and agreement testing to achieve arbitrarily small soundness error.
• These methods have significant applications in probabilistically checkable proofs and high-dimensional expanders, enabling robust agreement testing in complex combinatorial and algebraic settings.

Sparse direct-product testers are randomized algorithms designed to efficiently test whether a function (typically defined on faces of a combinatorial structure such as a simplicial complex or a product domain) is close to a direct-product encoding, using a constant or sparse number of queries and achieving soundness error that can be made arbitrarily small. These mechanisms have become a central tool in probabilistically checkable proofs (PCP) constructions and the paper of high-dimensional expanders, leveraging novel characterizations in terms of coboundary expansion and agreement testing. The term "sparse" in this context refers to the property that the underlying testing structure—such as the hypergraphs or complexes on which the queries are evaluated—has bounded or constant degree, rather than being globally dense.

1. Formal Setting and Definitions

Let $X$ be a $d$-dimensional simplicial complex with a finite vertex set $X(1)$ and a family of $k$-faces $$X(k) \subseteq \{\sigma \subseteq X(1): |\sigma|=k\}$$, downward-closed and ordered by inclusion. The maximum vertex degree $\Delta(X)$ is defined by

$$\Delta(X) = \max_{x \in X(1)} | \{ \sigma \in X(d): x \in \sigma \} |,$$

and $X$ is of constant degree if $\Delta(X) \leq \Delta$ for some fixed $\Delta$.

A function $F: X(k) \to \{0,1\}^k$ is a $k$-wise direct-product encoding of a function $f: X(1) \to \{0,1\}$ if for every ordered $k$-face $\sigma = \{x_1 < \cdots < x_k\} \in X(k)$,

$F(\sigma) = (f(x_1), \ldots, f(x_k)).$

The core testing problem is: given access to $F$, decide (with high probability from few queries) whether $F$ is correlated with a direct-product encoding.

Testers are randomized algorithms that make $T$ queries $\sigma^1, \ldots, \sigma^T \in X(k)$ to $F$ and decide acceptance or rejection based on the answers. Completeness is the guarantee that true encodings are always accepted; soundness is quantified as the largest $s$ such that any $F$ accepted with probability at least $s$ is $\epsilon$-correlated with a true encoding.

2. Sparse Tester Constructions: Canonical Designs

Sparse direct-product testers achieve constant query complexity and operate on bounded-degree (constant-degree) complexes or hypergraphs. For $\delta > 0$, the canonical tester of Bafna–Lifshitz–Minzer (Bafna et al., 1 Feb 2024) operates as follows:

1. Randomly sample a top-dimensional face $D \in X(d)$.
2. Choose a random subset $I \subset D$ of size $t = \lfloor \sqrt{k} \rfloor$.
3. Pick random $k$-faces $A, A' \subset D$, each containing $I$.
4. Query $F(A)$ and $F(A')$; accept if and only if $F(A)|_I = F(A')|_I$.

Completeness is immediate for perfect encodings. The key technical result establishes the existence of explicit $d$-dimensional complexes $X$ of constant degree and dimension such that for every $\delta > 0$, the above 2-query tester has soundness at most $\delta$ for all large enough $k$.

Beyond simplicial complexes, sparse direct-product testers extend to product domains and tensor networks. Dinur–Golubev (Dinur et al., 2019) present a “Square in a Cube” 4-query test for the direct-sum property, generalizing to a wide class of rank-1 tensor product testers.

3. Coboundary Expansion and Characterization

The correctness and soundness of sparse direct-product testers are intrinsically linked to high-dimensional expansion properties. The core notion is Unique Games (UG) coboundary expansion: for a non-Abelian group $G$ (often $S_m$), the coboundary operator $\delta$ acts on $G$-valued functions on faces, with e.g. for $k = 1$: $$(\delta \pi)(x, y, z) = \pi(x, y) \cdot \pi(y, z) \cdot \pi(z, x).$$ A complex is a $(m, r, \xi, c)$-UG-coboundary expander if, for any assignment of permutations (e.g. labelings) on the edge graph of its $r$-faces, low inconsistency (fraction $\leq \xi$ of failings on triangles) ensures closeness to a global labeling except on $O(c)$ fraction of edges.

Bafna–Minzer and separately Dikstein–Dinur established the equivalence (in spectral expander settings): a complex admits a 2-query direct-product tester with soundness $s_0$ if and only if it is a UG-coboundary expander with expansion parameters determined by $s_0$ (Bafna et al., 1 Feb 2024, O'Donnell et al., 13 Nov 2025). Formally, any function nearly satisfying cocycle conditions must be globally close (in $\ell_2$ norm) to a true cocycle.

4. Explicit Complexes and Technical Developments

The explicit construction of suitable complexes is nontrivial and central to sparseness and soundness:

• Chapman–Lubotzky Complexes. These are quotient complexes derived from Bruhat–Tits buildings of type Ċ$_n$ over $p$-adic fields, with arithmetic lattices in $\mathrm{Sp}_{2n}(\mathbb Q_p)$. All links are (products of) spherical buildings that provide $\epsilon$-product measures ($\epsilon \approx 1/\sqrt{p}$), facilitating local expansion. The proof of coboundary expansion employs an intricate “lopsided induction” on $r$: base cases involve established topological and spectral arguments; the inductive step leverages “list-itinerary” alignments, controlled by product measures, with coboundary constants growing only $2^{o(r)}$ (Bafna et al., 1 Feb 2024).
• Kaufman–Oppenheim Coset Complexes. These arise from coset designs over matrix groups $G = \mathrm{SL}_{n+1}(R)$, with staircase subgroups indexed to define partite structure (O'Donnell et al., 13 Nov 2025). The direct-product testing problem reduces to verifying local and global expansion (via r-triword—1-coboundary—expansion) in the local complexes, which is achieved by bounding Dehn functions in unipotent subgroups: specifically, efficient presentations yield simple connectivity and filling-area bounds required for coboundary expansion. These complexes are fully explicit, strongly regular, and elementary to describe (requiring none of the deep number theory underlying building quotients).

In both families, the blow-up in dimension $d$ and degree $\Delta$ is a function of $1/\delta$, but remains bounded for fixed $\delta$.

5. Agreement Testing and Soundness Theorems

The sparse direct-product testing regime is unified through the agreement-testing paradigm. The seminal theorem states: For each fixed (sufficiently large) $k$, there exists a constant-degree $d$-dimensional complex and a tester (querying two random $k$-faces $A, A'$ per top-face, conditioned on $|A \cap A'| \approx \sqrt{k}$) such that if acceptance probability $\geq \delta$, then $F$ is at least $\delta^{O(1)}$-correlated with some direct-product encoding (Bafna et al., 1 Feb 2024, O'Donnell et al., 13 Nov 2025).

In the case of KO coset complexes, the "V-Test" agreement tester samples hyperedges corresponding to random $k$-tuples in the complex's parts, querying two provers (on possibly overlapping sets) and accepting on agreement over the intersection. Soundness is established using local expansion (spectral and product), triviality of global $1$-cohomology (no nontrivial covers), and—critically—dimension-independent coboundary expansion in 2-dimensional links, shown via Dehn-presentation arguments.

6. Applications to PCPs and Broader Context

Sparse direct-product testers with low soundness underpin optimal PCP constructions. The existence of constant-query testers with arbitrarily small soundness parameter allows the construction of PCPs of quasilinear length and constant soundness, as implemented in Bafna–Minzer–Vyas BMVY25. The fully explicit nature and elementary description of KO complexes eliminate the necessity for deeper Bruhat–Tits and arithmetic lattice theory, marking a major structural simplification.

A summary comparison of leading explicit constructions:

Construction	Degree Bound	Explicitness	Expansion Technique
Chapman–Lubotzky Quotients	$O_\delta(1)$	Building quotient	Lopsided-induction
Kaufman–Oppenheim Complexes	$O_\delta(1)$	Coset, elementary	Dehn presentations

A plausible implication is the growing capacity for elementary, explicit, and efficient agreement-testing frameworks in high-dimensional combinatorics, with direct impact on the PCP theorem and related complexity-theoretic reductions.

7. Variants and Theoretical Guarantees

Several alternative testers are possible within the same agreement-testing formalism. For functions $$f: [n_1] \times \cdots \times [n_d] \to \mathbb F_2$$, Dinur–Golubev (Dinur et al., 2019) describe a 4-query “Square in a Cube” test and a $(d+2)$-query “Shapka” test for the direct-sum (rank-1 tensor) property. Both achieve completeness (direct sums pass with probability 1) and soundness with explicit constants independent of the dimension, relying on local-to-global stitching via agreement theorems and combinatorial arguments.

These developments confirm that, across a broad class of structures, sparse direct-product testers can robustly distinguish true product encodings from functions far from this property, using only a small number of local checks—with guarantees closely tied to high-dimensional expansion.

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References:

• (Bafna et al., 1 Feb 2024): M. Bafna, Z. Lifshitz, D. Minzer, "Constant Degree Direct Product Testers with Small Soundness"
• (O'Donnell et al., 13 Nov 2025): S. O'Donnell, N. Singer, "Low-soundness direct-product testers and PCPs from Kaufman--Oppenheim complexes"
• (Dinur et al., 2019): I. Dinur, E. Golubev, "Direct Sum Testing: The General Case"