Dimension Expanders: Theory and Applications

• Dimension expanders are collections of linear operators that increase the dimension of small subspaces by a prescribed factor, acting as a linear-algebraic analogue to graph expanders.
• They are constructed explicitly using methods from representation theory, coding theory, and monotone expander techniques, as well as via probabilistic approaches.
• Their applications span tensor rank lower bounds, coding theory, pseudorandomness, and complexity theory, driving advances in algebraic and computational research.

A dimension expander is a collection of linear operators on a finite-dimensional vector space such that, for every sufficiently small subspace, the summed images of these operators increase its dimension by a prescribed factor. This concept forms a central pillar in linear-algebraic pseudorandomness, providing a structural analogue of graph expanders within the field of linear maps, and directly impacting applications in tensor rank lower bounds, coding theory, pseudorandomness, and computational complexity.

1. Formal Definitions and Notions

Let $\mathbb{F}$ be any field (finite, $\mathbb{R}$, or $\mathbb{C}$), and let $n \in \mathbb{N}$. A family $\mathcal{A} = \{L_1,\dots,L_D\}$ of linear maps $L_i : \mathbb{F}^n \to \mathbb{F}^n$ (i.e., $n\times n$ matrices) is called a $\tau$-dimension expander if for every subspace $U \subseteq \mathbb{F}^n$ with $\dim U \leq n/2$,

$$\dim\Bigl( \sum_{i=1}^D L_i(U) \Bigr) \geq (1+\tau)\cdot\dim U.$$

Alternatively, for $d \in \mathbb{N}$ and $\alpha, \varepsilon > 0$, a $(d,\alpha,\varepsilon)$-dimension expander is a family $\{T_1,\ldots,T_d\}$ with

$$\forall W \leq \mathbb{F}^n,~\dim W \leq \alpha n \implies \dim\Bigl( W + \sum_{i=1}^d T_i(W) \Bigr) \geq (1+\varepsilon) \dim W.$$

A related notion is dimension-spreading: $\mathcal{A}$ is $(s,t)$-dimension-spreading if for every subspace $U \subseteq \mathbb{F}^n$ with $\dim U \geq s$,

$\dim\Bigl( \sum_{i=1}^D L_i(U) \Bigr) \geq t.$

Dimension expanders can be compared with linear-algebraic analogues of spectral and edge expansion found in quantum channels; these relationships form a strict hierarchy not present in graph expansion (Li et al., 2022).

2. Explicit and Probabilistic Constructions

Given their combinatorial and computational significance, explicit constructions of constant-degree dimension expanders are central to the theory.

a) Representation-Theoretic Methods:

Lubotzky–Zelmanov (2008) provided constructions over $\mathbb{C}$ using finite groups $G$ whose Cayley graphs are vertex-expanders. For a generating set $S$, the image $\rho(S)$ under any irreducible unitary representation $\rho: G \to U(\mathbb{F}^n)$ constitutes a $\tau$-dimension expander for some $\tau>0$ related to the spectral gap (Dvir, 4 Nov 2025).

b) Coding-Theoretic and Rank Condenser Approaches:

Fialkovski–Guruswami and Guruswami–Ramita–X. constructed dimension expanders via combinatorial design and subspace designs, obtaining $D=O(1)$ and $\tau = \Omega(1)$ over suitably large fields. Forbes–Guruswami (Forbes et al., 2014) introduced a construction based on tensoring and (lossy) rank condensers:

• Embed $x\in \mathbb{F}^n$ via $d$ different block-placements ($T_i$), boosting dimension by $d$.
• Use explicit families of linear maps $E_j$ (Wronskian rank condensers) to compress while retaining rank, composing to obtain expanders $A_{i,j}$.

This method yields explicit, degree-$O(1)$, expansion-$\Omega(1)$ expanders for $|\mathbb{F}|=\text{poly}(n)$.

c) Monotone Expander Based:

Bourgain–Yehudayoff established the only known construction over all fields $\mathbb{F}$ with constant $D$ and $\tau$, via constant-degree monotone bipartite expanders. Each monotone matching $f$ is converted into a partial map $f: [n] \to [n]$, with associated matrix $A_f$, to form $\tau$-dimension expanders (Dvir, 4 Nov 2025).

d) Probabilistic Methods:

Random constant-size tuples of permutation or unitary matrices yield $(n,d,\mu)$-dimension expanders with high probability for some absolute $\mu>0$ (Li et al., 2022).

Construction Method	Degree $D$	Field	Explicitness
Representation-theoretic	Constant	$\mathbb{C}$	Explicit (via groups)
Coding-theoretic	$O(1)$	Large $\mathbb{F}$	Explicit (Wronskian/rank codes)
Monotone expander	Constant	Arbitrary $\mathbb{F}$	Explicit
Probabilistic	Constant	Any	Probabilistic/existential

3. Hierarchy of Linear-Algebraic Expansion Notions

The theory of dimension expanders sits within a broader landscape of linear-algebraic expansion, exhibiting a strict hierarchy not mirrored in graph expansion.

• Dimension Expanders (vertex-analogue): defined as above.
• Dimension Edge Expanders: For $B = (B_i) \subseteq M(n,\mathbb{F})$,

$$h_D(B) = \min_{1 \leq \dim V \leq n/2} \frac{1}{d\dim V} \sum_{i=1}^d \mathrm{rank}(B_i|_{V^\perp, V}).$$

Equivalent to dimension expansion (Li et al., 2022).

• Quantum Expanders and Quantum Edge Expanders (spectral/edge analogues over $\mathbb{C}$): A tuple $\{B_i\}$ is a quantum expander if its associated quantum channel $\Phi_B(X) = \frac{1}{d} \sum B_i X B_i^*$ exhibits a spectral gap. Quantum edge expansion relates to the mixing of projections under $\Phi_B$.

The relationships are summarized as:

$$[\text{quantum expander}] \Longleftrightarrow [\text{quantum-edge expander}] \Longrightarrow [\text{dimension-edge expander}] \Longleftrightarrow [\text{dimension expander}]$$

However, none of the backward implications hold in general, i.e., there exist dimension expanders which are not quantum expanders (Li et al., 2022).

4. Connections to Tensor Rank and Lower Bounds

A principal application of constant-degree dimension expanders is the construction of explicit third-order tensors with near-maximal rank, influencing computational complexity and circuit lower bound theory.

Given a family $\mathcal{M} = \{A_1, \ldots, A_D\}$ of $n \times n$ matrices, form the third-order tensor

$T^{\mathcal{M}}(i, j, k) = (A_i)_{jk}$

with “slices” corresponding to the $A_i$. If $\mathcal{M}$ is $(s,t)$-dimension-spreading, the rank lower bound theorem states:

$\mathrm{R}(T^{\mathcal{M}}) \geq n + t - s$

and the same holds for border-rank over $\mathbb{R}$, $\mathbb{C}$ (Dvir, 4 Nov 2025).

By constructing $(n,(1-\varepsilon)n)$-spreading families of constant size, one obtains explicit tensors of shape $(D, n, n)$ and rank at least $(2-\varepsilon)n$. This improves prior explicit constant-mode constructions, whose best known bounds were $3n/2$.

This algebraic bridge underscores that advances in dimension expander constructions (smaller $D$, larger $\tau$) directly strengthen explicit tensor rank lower bounds and, by extension, arithmetic circuit lower bounds.

5. Algebraic and Geometric Existence via Quiver Theory

The existence and parameter boundaries for dimension expanders have been characterized using geometric and representation-theoretic methods.

By relating dimension expanders to generic representations of Kronecker quivers (two vertices, $m$ arrows), existence reduces to the non-vanishing of specific open sets in the representation space:

• The sharp existence criterion: for $m \geq 1$, rational slope $a = d_2/d_1$, density $0<\delta<1$, and expansion parameter $\varepsilon>0$, there exist $(\delta,\varepsilon)$-expander representations iff

$\varepsilon < E_m(a,\delta),$

where $E_m(a,\delta)$ is an explicit function derived from quiver geometry (Reineke, 2022).

The classical case, with $k$ maps on $N$-dimensional $V$, yields existence of $(k,\varepsilon)$-dimension expanders with $$\varepsilon \leq E_k := (k+1 - \sqrt{k^2 - 2k + 5})/2$$ for algebraically closed $F$.

This approach is conceptual and non-constructive, showing that generic collections satisfy expansion properties without explicit matrix descriptions.

6. Applications and Algorithmic Impact

Dimension expanders serve as algebraic analogues to graph expanders in derandomization and computational hardness:

• Coding Theory: Underpin list-decodable codes, especially with constructions derived from folded Wronskians and subspace designs (Forbes et al., 2014).
• Pseudorandomness: Enable the construction of affine extractors, derandomization in polynomial identity testing, and extractors for linear-algebraic sources.
• Complexity Theory: Feed into hardness versus randomness paradigms, impacting explicitness in algorithms for matroid intersection and solvability of linear systems.

There is a direct correspondence between explicit dimension expanders and explicit two-source rank condensers, or, through rank-metric code theory, with linear-algebraic condensers optimal up to constant factors.

7. Outlook and Open Problems

While significant progress has been made in explicit and generic constructions, important questions remain:

• Achieving optimal expansion parameters $\varepsilon$ with explicit constant-degree constructions over arbitrary fields remains open.
• Whether monotone expander constructions can be further simplified or generalized to smaller $D$ or larger $\tau$.
• The effect of improved expanders on pushing known explicit rank lower bounds for tensors and, consequently, arithmetic circuit lower bounds.

A plausible implication is that advances in dimension expander theory will yield new algebraic constructions for tensor complexity, extraction, and derandomization, feeding into broader questions in computational complexity and information theory.