Minimum Required Frame-Set (MRFS)

• MRFS is a formal construct defining the minimal set of frames, states, or time-frequency parameters essential for preserving and reconstructing critical content.
• It operationalizes minimal evidentiary demand, driving methodology in video coding, time-frequency analysis, and quantum gate characterization through practical minimality.
• MRFS informs applications from resource-efficient video encoding to rigorous quantum assessments, balancing theoretical sufficiency with experimental constraints.

The Minimum Required Frame-Set (MRFS) is a formal construct denoting the smallest collection of frames, input states, or time-frequency parameters necessary to preserve, reconstruct, or characterize essential content, properties, or functional performance of signals, systems, or tasks. MRFS principles span domains including time-frequency analysis, video coding, multi-frame question answering, and quantum device characterization, reflecting the interplay between theoretical minimality and practical sufficiency.

1. Foundational Definitions and Theoretical Formulation

The MRFS quantifies minimal evidentiary or informational demand in a task-dependent manner. In video understanding, MRFS denotes the smallest number $k$ such that, for a video $v$, question $q$, ground-truth answer $y$, model $f$, selector $r$, and maximum frame budget $x$, the model predicts the correct answer after fusing the top $k$ relevant frames:

$$\text{MRFS}_x(q;f,r) = \min_{1 \leq k \leq x} \big\{ k : E(f(q, \{\pi_1,\ldots,\pi_k\}), y) = 1 \big\}$$

with $E(\cdot, \cdot)$ as the accuracy indicator, and $\pi_1, \ldots, \pi_k$ the selector-preferred frames. In time-frequency analysis, MRFS relates to the smallest proven region $S \subset \mathbb{R}^2_+$ of sampling parameters for which windowed Gabor frames are guaranteed, reflecting both analytic and numerical sufficiency. For quantum gate characterization, MRFS is the minimal set of input states needed to guarantee injective distinguishability among process realizations.

2. MRFS in Time-Frequency Analysis: Gabor Frames for $B_2$

MRFS arises in determining frame sets of the second-order $B$-spline $B_2$ under Gabor frame constructions for $L^2(\mathbb{R})$ (Atindehou et al., 2018). Here, MRFS is the smallest explicitly described region $S \subset \mathbb{R}^2_+$ such that $G(B_2, a, b)$ forms a frame. Analytically, explicit regions are proved: $$\Lambda(B_2) \supset \Gamma_3 \cup \bigcup_{m=3}^\infty \Lambda_m$$ Where

• $$\Gamma_3 = \left\{ a \in (0, 2/9] \cup (2/7, 1/2),\, b \in \left(4/(2+3a),\, 2/(1+a)\right] \right\}$$
• $\Lambda_m$ comprises parameter intervals indexed by $m$, admitting explicit, compactly supported dual windows via matrix inversion and determinant positivity. MRFS is the union of such analytically validated regions; numerical results suggest further inclusion (the "purple diamond" gap), but absent a closed-form dual, these points do not formally constitute MRFS.

MRFS thus anchors lower bounds for provable Gabor frame parameter regions, steering rigorous window design and prompting further analytic investigation of currently unproven gaps.

3. MRFS in Green Video Coding: Critical Frame-Rate Discovery

In video broadcasting, MRFS is operationalized as the "critical frame-rate" $f_0$: the lowest in $\{30, 60, 120\}$ fps at which post-decimation video remains perceptually indistinguishable from original $120$ fps (Herrou et al., 2020). MRFS is determined via cascaded random forest classifiers operating on feature vectors extracted from 4-frame chunks. Frame-rate selection is formalized as learning a mapping $f:\mathbb{F} \to \{30, 60, 120\}$, with $\mathbb{F}$ comprising aggregate motion, difference, gradient, and luminance features.

Ground-truth MRFS labels are established via subjective continuous evaluation where the critical frame-rate corresponds to the lowest $f$ such that the majority of experts observe no perceptual loss:

• $f_0$ for each chunk is encoded as the MRFS label Classification metrics and DMOS (differential mean opinion score) statistically validate that MRFS-based VFR maintains subjective quality while yielding significant bitrate (average $-4.3\%$) and encoder/decoder complexity ($\sim28\%$) savings. MRFS enables formal tradeoffs between resource usage and perceptual fidelity in broadcast video systems.

4. MRFS in Video Question Answering: Evidential Demand Quantification

HERBench introduces MRFS as a rigorous metric for multi-evidence video question answering (Ben-Ami et al., 16 Dec 2025). For a given $q$, MRFS is the minimum number of distinct, non-redundant frames the model must jointly attend to in order to answer correctly. The HERBench protocol computes MRFS via selector-driven frame ranking and binary search, excluding text-only solvable questions.

In practice, HERBench's mean MRFS is 5.49, exceeding that of comparable benchmarks (e.g., NExT-QA: 2.61; MVBench: 3.52; LongVideoBench: 4.07), evidencing higher compositional demand. MRFS correlates inversely with model accuracy, revealing bottlenecks in both evidence retrieval (selecting the necessary frames) and fusion (integrating cues when provided). Ablation tests using oracle selectors isolate these deficits, as even optimal frame retrieval yields sub-50% overall accuracy, highlighting current Video-LLMs' limitations in compositional integration.

MRFS thus supplies quantifiable minimum evidence standards, directly informing the design of next-generation multi-frame fusion architectures.

5. MRFS in Quantum Gate Characterization: Informational Minimality

For quantum process characterization, MRFS is the size of the smallest set of input states (a frame-set) guaranteeing algebraic distinguishability among unitaries and enabling non-unitarity detection (Reich et al., 2013). Standard quantum process tomography requires $d^2$ inputs for Hilbert space dimension $d$. Algebraic analysis reveals:

• Two-state MRFS $(\rho_1=I/d,\, \rho_2=|\psi\rangle\langle\psi|)$ is information-theoretically sufficient for injectivity, but impractical due to preparation constraints and absence of direct fidelity bounds.
• $(d+1)$-state MRFS (computational basis $+\,$ totally rotated state) admits numerically tight fidelity estimates, yielding $F_\lambda$ with empirically validated bounds on $F_{\mathrm{avg}}$.
• $2d$-state MRFS (two mutually unbiased bases) enables analytical Hofmann bounds, establishing both lower and upper fidelity limits via measured classical fidelities $F_1$, $F_2$.

The choice of frame-set—balancing size, generation ease, and bound tightness—directly impacts measurement overhead, state preparation complexity, and experimental scalability. MRFS thus operationalizes minimal sampling sufficient for full gate fidelity characterization and non-unitarity testing.

6. Practical Applications, Limitations, and Extensions

MRFS formalism guides:

• Signal and window design in time-frequency analysis (provable frame regions, explicit dual construction).
• Classification-driven VFR in video coding, with direct implications for hardware complexity and bandwidth minimization.
• Benchmark creation in VideoQA, exposing granularity of compositional integration and retrieval bottlenecks.
• Experimental protocol and resource allocation in quantum device assessment.

Limitations arise where theoretical MRFS is difficult to realize experimentally (e.g., maximally mixed states for quantum gates), or where current models cannot attain the implied integration standard (e.g., Video-LLMs with high MRFS tasks). Extensions include retraining to alternate frame-rate sets (broadcast standards), benchmarking with varying frame budgets ($x$), and design of domain-specific selectors or fusion schemes.

MRFS, across domains, remains pivotal in connecting minimal sufficient sampling, system characterization, and compositional reasoning—each aspect subject to ongoing research refinement.