Single Vector Paradigm

• The Single Vector Paradigm is a framework that uses a single vector entity to describe and generate rich observables across fields such as cosmology, quantum field theory, and computational science.
• It employs methodologies from vector field measurements, duality transformations in field theory, and matrix analysis to achieve efficient observer design, signal reconstruction, and hardware optimization.
• This paradigm unifies scalar and vector operations in algebraic structures and reduces system complexity, leading to practical benefits in simulation accuracy and computational performance.

The Single Vector Paradigm encompasses a range of theoretical, algorithmic, physical, and computational concepts grounded in the idea that a single vector entity—whether a field, measurement, or algebraic object—suffices to describe, reconstruct, or generate rich physical observables or computational optimizations. In technical contexts, this paradigm is invoked for: (i) the generation of cosmological curvature perturbations from a single vector field, (ii) the unified formulation of three-dimensional quantum field theories via single excitation vector models, (iii) single-measurement-based system identification and observer design, (iv) algebraic frameworks where vector and scalar attributes are unified, (v) compressed matrix property tests and signal reconstructions using only a single vector, (vi) efficient hardware/software mapping by leveraging vectorized computation as an abstraction, and (vii) sparse signal processing using reconstructed vector signals from a single sensor. The paradigm thus manifests across diverse subfields, consistently privileging the sufficiency and centrality of single-vector quantities.

1. Vector Curvaton Cosmology and Statistical Anisotropy

The vector curvaton mechanism exemplifies the Single Vector Paradigm in early universe cosmology by positing that a single massive Abelian vector field—distinct from the inflaton—can be responsible for generating or contributing to the primordial curvature perturbation, ζ. During inflation, the vector field, not responsible for driving expansion, acquires nearly scale-invariant superhorizon perturbations. When the effective mass of the field (m) becomes comparable to or larger than the Hubble rate, coherent oscillations begin; the energy density then redshifts as pressureless matter ($\propto a^{-3}$) and its zero mode’s anisotropic stress averages out over a Hubble timescale.

The contribution of the vector field to ζ is governed by:

$$\zeta = N_\phi\, \delta\phi + N_A\, \delta A + \frac{1}{2} N_{AA} (\delta A)^2 + \ldots$$

The statistical anisotropy this induces manifests in both the power spectrum,

$$P_\zeta(\mathbf{k}) = P_\zeta^{\mathrm{iso}}(k) \left[1 + g\, (\hat{d} \cdot \hat{k})^2 + \ldots\right],$$

and the bispectrum, with the parameters $g$ and $G$ quantifying the degree of anisotropy and angular modulation, directly linked to the preferred direction established by the vector field. The model predicts correlated signatures of anisotropy in both the CMB power spectrum and its bispectrum, serving as distinct observational fingerprints. Two specific realizations are analyzed: one with non-minimal gravitational coupling (inducing ~100% anisotropic particle production) and a second with a time-dependent kinetic function and mass, which—for sufficiently large mass—yields an isotropic regime enabling the vector field to generate all of ζ (Dimopoulos, 2011).

2. Single Excitation Vector Models in 3D Field Theory

In three-dimensional quantum field theory, the Single Vector Paradigm is realized in master actions that interpolate between distinct, yet canonically equivalent, massive vector models—specifically, self-dual and topologically massive models. This is achieved through a sequence of duality transformations, each step preserving the propagation of a single massive excitation.

A generic unified action takes the form:

$$I[a,\,A] = \langle -\frac{m}{2} a^2 + A \cdot \epsilon \partial a \rangle$$

and, depending on which field is eliminated, yields either the self-dual or topologically massive form. Hamiltonian (2+1) analysis, with a decomposition of fields into gauge-invariant transverse and longitudinal parts, confirms the equivalence by explicit reduction to single-mode propagation. Non-local actions, appearing naturally via the path integral after integrating out auxiliary fields, remain physically consistent and free of ghosts provided proper gauge-fixing is employed. This unified view allows further extension to non-local models and offers practical routes to address higher-derivative (potentially ghost-ridden) theories, bosonization in 3D, and even higher-spin or gravity analogues (Khoudeir et al., 2011).

3. Single-Vector Characterization in Matrix Theory

Matrix analysis under the Single Vector Paradigm focuses on detecting global properties (such as total negativity) using tests involving only a single carefully chosen vector. For a matrix $A$, total negativity (all minors of order ≤$k$ negative) is characterized by three equivalent criteria:

• Variation Diminishing Property: For vectors $x$ in the alternating bi-orthant,

$S_+(Ax) \leq S_-(x),$

where $S_+$ counts sign changes without zeros.

• Sign Non-Reversal Property: There exists at least one coordinate $i$ for which $x_i (Ax)_i > 0$.
• LCP Characterization: The Linear Complementarity Problem has solutions restricted to $\{0, -x_{A_r}\}$ for special $x_{A_r}$ constructed from adjugates and alternating-sign vectors.

Applied to the interval hull problem, this means that only two “corner” matrices in the interval must be checked, drastically reducing complexity by leveraging single-vector sufficiency (Choudhury, 2021).

4. Signal and Observer Design: Single-Vector Measurements

The Single Vector Paradigm supports advanced observer/controller design where estimation of a dynamic state (e.g., angular velocity) is performed directly from the time history of a single vector measurement, bypassing the need for full-system state reconstruction. In rigid body dynamics, angular velocity $\omega$ can be inferred from the body-frame measurement of a single inertial vector $a(t)$ using a nonlinear observer:

$$\dot{\hat{X}} = \begin{bmatrix} a \times \hat{a} - k(\hat{a} - a) \ E(\hat{\omega}) + J^{-1}\tau + k^2 a \times (\hat{a} - a) \end{bmatrix}$$

where $E(\cdot)$ encodes the quadratic part of the Euler equations, and $k$ is a tunable gain. Convergence is proven under a persistence of excitation (PE) condition on $a(t)$, which, for free-rotating bodies, is generically satisfied except in degenerate cases. The paradigm thus both reduces sensor requirements and enables robust observer synthesis (Magnis et al., 2015).

5. Unification of Scalar and Vector Quantities in Algebraic Structures

Algebraic frameworks such as paravectors extend the single vector concept by encapsulating scalar and vector components in a unified mathematical object:

$$\Gamma = \begin{bmatrix} \alpha \ \beta \end{bmatrix}, \quad \alpha \in \mathbb{C},\; \beta \in \mathbb{C}^3,$$

with multiplication defined to generalize both the dot and cross products. The algebra is non-commutative but associative, forming a ring with unity and supporting operations reflecting geometric notions of parallelism, perpendicularity, and angle (with components such as “cosi Φ” and “dextis Φ”). Key geometric and algebraic theorems (parallelogram law, polarization identity) are preserved in this structure. This unification yields a powerful, single-entity approach for modeling relativistic four-vectors, transformations, and operations—facilitating deeper connections between algebra, geometry, and physics (Józef, 2016).

6. Sparse Signal Processing and Sensor Applications

In sensor processing, especially direction-of-arrival (DOA) estimation with single vector hydrophones, the Single Vector Paradigm manifests through signal model reconstruction. The technique transforms the multi-channel covariance matrix from a single sensor into a Toeplitz matrix via vector signal reconstruction (VSR) using an appropriate transformation matrix $G$. This enables the direct application of gridless sparse algorithms (e.g., Atomic Norm Minimization, Structured Covariance Estimation, Sparse and Parametric Approach) originally tailored for uniform arrays. The process involves, for received signal $x(t)$,

$y(t) = G x(t),$

with redefined covariance and steering vectors. The gridless approaches, following VSR, optimize DOA estimates via convex or semidefinite programming, outperforming traditional methods under low SNR and multi-source conditions—demonstrated by simulation showing greater resolution probability and lower estimation bias (Guo, 21 Apr 2024, Guo, 21 Apr 2024).

7. Computational and Hardware Realizations

The high-level programming, compilation, and hardware architectures discussed in this paradigm exploit vectorization by treating all program data as vectors at the semantic and system level. Examples include:

• cphVB: Provides a transparent pathway from high-level vectorized code (e.g., NumPy in Python) to optimized, architecture-agnostic vector bytecode. All computations are handled at the vector (array) level, enhancing both abstraction and performance; speedups (up to 6.8× for kNN) validate the efficacy (Kristensen et al., 2012).
• Univec: Applies compile-time dimensional checking to vector, matrix, and generalized (complex/quaternion) operations, treating all as dimensionally resolved vectors. Template-based type safety and broad operator support ensure correctness and performance competitive with established libraries (Stan et al., 2022).
• Ara2 Processor: Implements an RVV 1.0 compliant open-source vector processor where the single vector register file is managed as a globally contiguous, lane-distributed structure, supporting high functional-unit utilization (~95%) for vector workloads. The trade-offs between single-core (long vectors) and multi-core (multiple short vectors) configurations are explicitly quantified, with energy efficiency measured at 37.8 DP-GFLOPS/W for the 2-lane configuration; performance scaling and bottlenecks (e.g., in scalar core issuing and bank conflicts) are analyzed in detail (Perotti et al., 2023).

Conclusion

Across cosmology, field theory, linear algebra, estimation theory, algebraic systems, signal processing, and computational engineering, the Single Vector Paradigm underpins frameworks wherein a single vector object—be it a physical field, measurement, algebraic composite, or computational data structure—characterizes, generates, or reconstructs the information or observables of an entire system. The paradigm is validated in both theoretical rigor (cosmological perturbations, master action dualities, algebraic theorems) and empirical or computational performance (efficient DOA estimation, accelerated software/hardware, robust observer design), with constraints, limitations, and avenues for future research precisely delineated in each domain.