Simultaneous Points of Minimal Sensitivity

• Simultaneous points of minimal sensitivity are well-defined configurations that isolate critical responses to perturbations across diverse mathematical and applied domains.
• They enable efficient sensitivity analysis by reducing complex systems to minimal identifiable sets, enhancing algorithmic convergence and stability.
• These concepts find practical applications in optimization, Boolean complexity, dynamical systems, and machine learning to improve robustness and inference accuracy.

Simultaneous points of minimal sensitivity arise across multiple mathematical and applied domains, notably optimization, Boolean and hypergraph complexity, dynamical systems, and deterministic inference under perturbation. This concept encompasses settings where a well-chosen subset or configuration of points captures all essential sensitivity properties—either in response to perturbations, in encoding critical solutions, or in the manifestation of system robustness—thereby enabling rigorous analysis and algorithmic exploitation of local or global invariance structures.

1. Minimal Sensitivity and Identifiable Sets in Variational Analysis

Within nonsmooth and variational optimization, particularly as formalized by set-valued mappings and subdifferential calculus, an identifiable set at $(\bar{x},\bar{v})$ for a set-valued mapping $G:\mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is any subset $M \subset \mathbb{R}^n$ satisfying: every sequence $(x_i,v_i)\to(\bar{x},\bar{v})$ with $v_i\in G(x_i)$ eventually has $x_i \in M$ for large $i$ (Drusvyatskiy et al., 2012). The (locally) minimal identifiable set $M$—unique whenever it exists—encapsulates the locus where criticality is robust under small problem perturbations.

Minimal sensitivity is thus realized by confining the analysis to $M$, since all approximate solutions to small perturbations reside within it:

• For (perturbed) problems $\min f(x) - \langle w, x\rangle$, the collection of all local minimizers for small $w$ is exactly $M$ (locally).
• Once $M$ is identified, optimality and sensitivity analysis reduce to behaviors on $M$ alone; for instance, quadratic growth conditions for $f$ globally are equivalent to those for $f|_M$.
• In active-set and algorithmic frameworks, finite identification of $M$ allows subsequent convergence to proceed on a (simpler, possibly smooth) reduced space.

Formally, minimal sensitivity is encoded by $M = G^{-1}(V)$, where $V$ is any neighborhood of $\bar{v}$, and all nearby perturbed stationary points $x$ with $0\in \partial f(x)-w$, $w\sim 0$, are in $M$ [cf. key formula in (Drusvyatskiy et al., 2012)].

2. Minimal Sensitivity in Boolean and Hypergraph Function Complexity

In complexity theory, minimal sensitivity of a Boolean function $f$ on $n$ variables is the minimum, over all inputs $X$, of the sensitivity $Sen(f, X)$—the number of bit-flips altering $f(X)$ (Li et al., 2015). For certain combinatorial functions, such as the simplified weighted sum function $f(X) = x_{s(X)}$, the behavior of minimal sensitivity is intimately tied to arithmetic properties of $n$, especially primality:

• For $n = p > 4$ (prime), $\min S(f) = 1$: every input is sensitive to some single bit-flip (Li et al., 2015).
• For composite $n$, this minimum can be 0 or 2, revealing the function's robustness or extreme fragility.
• This observation enables minimal sensitivity to serve as a potential primality indicator: deviation from 1 implies non-primality.

In the case of $k$-uniform hypergraph properties, the simultaneous occurrence of points of minimal sensitivity is embodied by inputs where flipping a single edge changes the property, while the function admits a quadratic gap to block sensitivity (Biderman et al., 2015). The construction ensures the set of sensitive points—minimally robust inputs—occurs in a highly structured, simultaneous way across the input domain.

3. Dynamical Systems: Multi-Sensitivity, Mean Sensitivity, and Factor Structure

Simultaneous points of minimal sensitivity in dynamical systems are rigorously captured by notions such as multi-sensitivity, $n$-sensitivity, and mean-sensitive tuples, and are deeply linked to the fiber structure over equicontinuous or distal factors.

For a continuous surjective selfmap $T:X\to X$ on a compact metric space:

• Multi-sensitivity: there exists $\delta>0$ so that, for any finite collection of open sets, a common time $n$ exists where each set demonstrates sensitive separation (Huang et al., 2015).
• Thick $n$-sensitivity and extensions for minimal group actions: for every open $U$ there exist $n$ points whose orbits are simultaneously separated by at least $\delta$ over a thick (or more structured) set of indices (Li et al., 2020, Li et al., 2021).
• These sensitivity properties are characterized via the size of the regionally proximal relation or the structure of the maximal equicontinuous or distal factor:

$(X, G) \text{ is $n$-sensitive} \iff \exists x\in X\ \#Q[x]\geq n$

where $Q[x]$ is the regionally proximal fiber over $x$ (Li et al., 2020, Li et al., 2021).

Mean-sensitivity and independence tuples extend this analysis to actions by amenable groups: in minimal systems with suitable factor and measure-theoretic regularity, every weakly mean-sensitive $K$-tuple is an IT-tuple (i.e., supports a rich independence structure) (Liu et al., 26 Jan 2025).

The following table summarizes dynamical system sensitivity phenomena:

Sensitivity Type	Characterization	Factor/Fiber Criterion
Multi-/Thick-$n$	$n$ separated points, thick set	fiber over equicontinuous max.
Mean-sensitivity	high-density $K$-tuples	regularity and IT-tuple fiber

In all these cases, the existence of such tuples/fibers—often maximal or thick—ensures a simultaneous, structural form of minimal sensitivity across the space.

4. Simultaneous Sensitivity in Optimization under Perturbations

The stability of strong minima under perturbation is central in variational analysis (Topalova et al., 2023). For a sequence of lower semicontinuous functions $f_n:X\to\mathbb{R}\cup\{+\infty\}$ converging pointwise to $f$ (subject to regularity and uniformity of level set convergence), there exists a dense $G_\delta$ set in a Banach perturbation space $P$ so that for each $g\in P$, every $f_n+g$ (and $f+g$) attains a strong minimum at $x_n$ (resp. $\hat{x}$), with $x_n\to\hat{x}$ as $n\to\infty$.

This construction achieves simultaneous points of minimal sensitivity: for all $n$, the minimizers $x_n$ are robust (the diameter of level-sets tends to zero) and converge under vanishingly small perturbations. The method applies equally to regularization, constrained problems, and stability analysis in the absence of convexity or uniform convergence (Topalova et al., 2023).

5. Complementary Perspectives: Reliability, State Estimation, and Non-Hermitian Systems

In reliability engineering, simultaneous estimation of complementary moment-independent sensitivity measures (the target index and the conditional index) allows identification of inputs that have minimal effect both on the global failure event and conditionally within the failure domain (Derennes et al., 2019). These indices, efficiently estimated via adaptive SMC and maximum entropy density estimation, quantify minimal sensitivity as an empirical property: the ability to detect input parameters for which the system is least responsive either globally or conditionally.

In observer design and process monitoring, simultaneous state and parameter estimation under limited observability employs sensitivity matrices to select estimable variables, leveraging their maximal collective information content (Liu et al., 2020). The selection algorithm orthogonalizes and ranks variables by sensitivity, freezing those with minimal contribution, thereby ensuring that the simultaneous estimation proceeds over a robustly sensitive subset leading to superior estimation accuracy.

Non-Hermitian physics illustrates another manifestation: tuning system parameters near higher-order exceptional points (EPs) enables a controlled transition in the sensitivity of eigenvalue splitting with respect to perturbations, effectively creating regimes where the response to input is minimal (or maximally enhanced), and the location of these sensitivity transitions is precisely engineered (Sahoo et al., 2022).

6. Simultaneous Minimal Sensitivity in Geometric and Machine Learning Settings

In geometric optimization, especially the design of minimization diagrams (including Voronoi diagrams), simultaneous points of minimal sensitivity are formalized by explicit sensitivity formulas for vertices where multiple cells intersect (Birgin et al., 2021). For example, the velocity vector $z_v'(0)$ at a vertex $v$ (intersection of three cells) is computed via the derivatives of the defining functions with respect to perturbations of the sites:

$$z_v'(0) = M(j, k, i) a_i + M(k, i, j) a_j + M(i, j, k) a_k$$

with $M(i,j,k)$ and related structure constants as functions of site positions. Optimization problems seeking configurations with prescribed minimal sensitivities exploit these formulas to produce diagrams (cellular decompositions) that are robust under (possibly simultaneous) small perturbations of the sites.

In explainable deep learning, sensitivity of convolutional neural network activations to simultaneous input augmentations is decomposed via Sobol indices and Shapley values (Kharyuk et al., 5 Mar 2025). These variance-based indices assign attributions to each augmentation (or their combinations), producing sensitivity maps that reveal which units or spatial regions are most/least affected by various perturbations. The single-class sensitivity analysis further identifies output classes minimally impacted by targeted activation masking—directly assessing simultaneous minimal sensitivity in the network’s inference chain.

7. Synthesis: Structural and Algorithmic Consequences

Across all these frameworks, the unifying theme is the identification or design of structures—be they subsets, tuples, fibers, or configurations—where the essential behavior under perturbations, changes, or uncertainty is concentrated. The minimality is both a mathematical and algorithmic guarantee: it ensures sufficiency (every nearby solution or sensitive response is accounted for) and efficiency (one need not consider extraneous variables or directions).

These simultaneous points or sets of minimal sensitivity have numerous algorithmic applications:

• Active-set and reducible-structure methods in optimization
• Minimal sensitivity points as robustness or primality indicators
• Construction of extremal functions for complexity separation (e.g., sensitivity vs. block sensitivity)
• Development of efficient and stable estimation or inference schemes under partial observability or ill-posedness
• Tuning of physical or learned systems to achieve desired balance between robustness and sensitivity

The connection with factor structure in dynamical systems, combinatorial independence in topological dynamics, and robust signal processing/decoding in probabilistic and geometric methods indicates the breadth and cross-disciplinary relevance of this variational-analytic perspective on minimal sensitivity.

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For precise definitions, structural theorems, and algorithmic details, see (Drusvyatskiy et al., 2012, Li et al., 2015, Huang et al., 2015, Biderman et al., 2015, Derennes et al., 2019, Liu et al., 2020, Li et al., 2020, Ricceri, 2020, Li et al., 2021, Birgin et al., 2021, Sahoo et al., 2022, Topalova et al., 2023, Portela, 20 May 2024, Liu et al., 26 Jan 2025), and (Kharyuk et al., 5 Mar 2025).