Virtual Linear Representation

• Virtual linear representation is a formal framework that transforms non-linear group actions into linear operations using reductions, quotients, or abstractions.
• It integrates domain-specific constructions from abstract algebra, quantum algorithms, machine learning, and control theory to achieve computational tractability and enhanced system analysis.
• Applications span improved concept encoding in representation learning, robust quantum state estimation with operator averaging, and stabilization of nonholonomic systems using invariant feedback control.

A virtual linear representation is a formalization of a group action (or, more generally, a concept direction or transformation) into linear (matrix or operator) actions on an associated module or vector space, often through an intermediate structure, reduction, or abstraction that is not present in the original system. This notion occurs in multiple fields: abstract algebra, quantum information, control theory, and machine learning—all with domain-specific constructions. Typically, it encapsulates actions that become linear only after passing to a quotient, submodule, canonical space, or by post-processing measurement data. Virtuality often signals that the representation is not strictly faithful or is defined only after reduction, projection, or partial inversion.

1. Group-Theoretic Virtual Linear Representations

In the theory of virtual Artin groups, the virtual linear representation generalizes the classical Krammer–Cohen–Wales representation to a broader context linking the Artin and Coxeter structures. For a Coxeter graph $\Gamma$, the virtual Artin group $VA[\Gamma]$ is generated by the Artin group $A[\Gamma]$ and the Coxeter group $W[\Gamma]$, intermixed by relations mimicking $W[\Gamma]$’s action on the root system $\Phi[\Gamma]$. The construction defines a module $V=\bigoplus_{\beta\in\Phi^+[\Gamma]}R\cdot e_\beta$, where $R=\mathbb{Z}[x^{\pm 1}, y^{\pm 1}]$ is a Laurent polynomial ring in two indeterminates (Dhanwani et al., 2024).

Linear representations are given for generators as follows:

• Artin generator: $\rho(O_s)(e_\beta) = e_{p(s)(\beta)}$ for $\beta\neq\alpha_s$; $x\,e_{-\alpha_s} + y\,e_{\alpha_s}$ for $\beta=\alpha_s$.
• Coxeter generator: $$\rho(T_s)(e_\beta) = \varepsilon(s, \beta)e_{p(s)(\beta)}$$ with sign determined by the bilinear pairing.
• Extension: For $w\in W[\Gamma]$, $\rho(w)(e_\beta) = y^{k(w, \beta)}e_{p(w)(\beta)}$ or $e_{p(w)(\beta)}$, depending on combinatorics.

These representations intertwine Artin-type and Coxeter-type blocks, resulting in a virtual version in the sense that the action is only strictly faithful on the quotient $VA[\Gamma]/PVA[\Gamma]'$, yielding a crystallographic group of rank $n=|\Phi[\Gamma]|$ with holonomy $W[\Gamma]$. When restricted to $A[\Gamma]$, the construction recovers the irreducible Cohen–Wales–Krammer module for spherical types. Virtual faithfulness is achieved by factoring out the pure-virtual commutator subgroup. In right-angled cases, this provides a route to proving residual finiteness and the $R_\infty$-property for automorphisms (Dhanwani et al., 2024).

2. Virtual Linear Concepts in Representation Learning

In the context of representation spaces for LLMs, virtual linear representation posits that high-level binary concepts (e.g., gender, truthfulness) are canonically realized as unit-length directions $\ell$ in an abstract representation space $\mathcal{C}$ (Nguyen et al., 22 Feb 2025). Here, virtuality corresponds to the statistical estimand computed from activation differences, mapped via invertible linear maps $\Psi$ (identity or whitening) and $\Psi^{-1}$ between activation space and $\mathcal{C}$.

The SAND (Sum of Activation-base Normalized Difference) procedure estimates $\ell$ by collecting difference activations from $k$ contrasting prompt pairs, forming normalized samples interpreted as draws from a von Mises–Fisher distribution, and then applying maximum likelihood estimation. The concept direction is

$$\lambda = \Psi^{-1}\left(\frac{1}{\|S\|_2} S\right),\quad S = \sum_{i=1}^k \frac{\Psi(\tilde{v}_i)}{\|\Psi(\tilde{v}_i)\|_2}$$

The result is a steering vector in activation space that “virtually” encodes the binary concept: no explicit token-level correspondence or unembedding inversion required.

Empirical validation demonstrates enhanced monitoring and intervention capabilities in LLaMA models, improved accuracy over PCA baselines, and robust performance across diverse concept manipulation tasks (Nguyen et al., 22 Feb 2025). The SAND approach expands the linear representation hypothesis beyond prior counterfactual-token protocols.

3. Virtual Linear Maps in Quantum Information

The Virtual Linear Map Algorithm (VILMA) realizes virtual linear representation by enabling operator averaging and variational optimization over the image of a reference quantum state $\rho_0$ under arbitrary circuit-like compositions of $k$-local maps, $\Lambda$, not necessarily physical or completely positive (García-Pérez et al., 2022). Measurement is performed via single-qubit informationally complete POVMs, yielding dual operators $\{D_m\}$ that act as classical shadows.

For any observable $O = \sum_k c_k P_k$ (Pauli decomposition), the estimator for expectation on the virtual image state $\rho_v = \Lambda(\rho_0)$ is

$$\bar{O}_\Lambda = \frac{1}{S} \sum_{s=1}^S \omega_{m^{(s)}},\quad \omega_m = \sum_k c_k\,\mathrm{Tr}[\Lambda(D_m) P_k]$$

Evaluation is executed by sweeping through the low-depth decomposition of $\Lambda$ and performing local operations and partial traces along causal cones for efficiency. Variational optimization is performed by expressing the estimator as affine in the Choi matrix $J(\Lambda_s)$ of each local map, allowing parameter updates via semidefinite programming.

Formal resource scaling is polynomial in $N$ for constant $k$ and depth, with main limitations due to exponential scaling when increasing circuit depth or local operator width. Proof-of-principle studies on molecular ground-state search and XX spin model demonstrate unbiased estimation and adaptive recovery of ground states in regimes accessible to classical shadow methods (García-Pérez et al., 2022).

4. Virtual Linear Nonholonomic Constraints in Mechanical Control

Virtual linear representations in control refer to invariant linear subbundles $D\subset TQ$ of the tangent bundle on a configuration manifold $Q$ defined by constraints $A(q)\dot{q}=0$, or $\mu^b(q)(v_q)=0$ for a set of one-forms $\mu^b\in\Omega^1(Q)$. The constraints are termed ‘virtual’ since they are enforced by feedback rather than by physical interactions or reaction forces (Simoes et al., 2024).

The existence of an exponentially stabilizing feedback $u(q,\dot{q})$ requires:

• Transversality: $D(q)\oplus F(q)=T_qQ$ for all $q$, with $F(q)=\mathrm{span}\{Y^a(q)\}$ being the input distribution.
• Full rank of $\{\mu^b\}$.
• Invertibility of the decoupling matrix $C^{ab}(q)=\mu^b(Y^a)(q)$.

The explicit stabilizing control is

$$u^*_a(q,\dot{q}) = C_{ab}(q)[-h^b(q,\dot{q}) - G h^b(q,\dot{q})]$$

where $h^b=\mu^b(q)\dot{q}$ and $G$ is the geodesic spray. This ensures exponential convergence of the system trajectory to the constraint manifold $D$.

Virtuality reflects the absence of physical enforcement: control is designed so that trajectories remain invariant in $D$, but unlike physical nonholonomic systems, there are no real constraint forces. The reduction on $D$ recovers exactly the dynamics of a true nonholonomic system (Simoes et al., 2024).

5. Embedding Nonlinear Systems via Virtual Linear Parameter-Varying (LPV) Representation

LPV representations embed nonlinear multiple-input multiple-output (MIMO) systems, originally characterized by linear fractional representations (NLFR), into affine linear state-space forms using automated factorization of static nonlinearities. Given an NLFR form, static nonlinearities $f(z)$ are decomposed as $f(z) = \bar{f}(z)z + c$ with scheduling functions $\bar{f}_i(z_1, ..., z_i)$ constructed recursively to avoid singularity (Schoukens et al., 2018).

A typical state-space LPV realization after offset removal and nonlinearity factorization reads:

$$\dot{x} = [A + B_w p(t) C_z]x + [B_u + B_w p(t) D_{zu}]u$$

$$y = [C_y + D_{yw} p(t) C_z]x + [D_{yu} + D_{yw} p(t) D_{zu}]u$$

where the scheduling variable $p(t)$ is given by the explicit analytic form of $\bar{f}$ evaluated on internal signals (e.g. positions, velocities).

Empirical verification on a 2-DOF mass–spring–damper system demonstrates that the LPV-embedded model perfectly matches the original nonlinear simulation. The approach supports exact embedding under conditions of strict properness, offset invertibility, and regularity of the nonlinearity factorization.

6. Interpretation and Domain-Specific Convergence

Across domains, ‘virtual’ linear representation is unified by the imposition of linear structure on objects or group actions that are not inherently linear but are rendered so after abstraction or quotienting. Implementations in group theory, representation learning, quantum algorithms, control theory, and nonlinear systems all exploit domain-specific mechanisms—quotients, canonical spaces, local mappings, or feedback laws. The central theme is computational or conceptual tractability achieved via virtualization, often yielding new faithfulness, invariance, or optimization properties not present in the original system.

A plausible implication is that future methods may generalize these virtual constructions to more complex nonlinearities, non-CP transformations, or ambiguous representation settings, always seeking a canonical or computationally efficient linearization. This suggests virtual linear representation will remain foundational for extending linear techniques into broader domains with intrinsic complexity.