Phase Alignability: Concepts & Applications

• Phase alignability is a structural property that measures the compatibility between observed phase configurations and a restricted set of transformations.
• It underpins applications in functional data analysis, synchronization, optical phase retrieval, and control systems by enabling reliable phase compensation and recovery.
• Analyzing phase alignability helps diagnose issues like gauge ambiguity and overfitting, guiding improvements in estimation methods and system robustness.

Searching arXiv for recent and directly relevant papers on phase alignability and related formulations. I’ll look up arXiv papers that use or define “phase alignability” across domains so the article can be grounded in the current literature. Phase alignability denotes the extent to which phase degrees of freedom can be represented, estimated, or compensated so that observations become consistent with a shared model. In current arXiv literature, the term is not attached to a single formalism: in functional data analysis it concerns whether smooth monotone warps can remove lateral variability without distorting amplitude; in synchronization it concerns recovery of latent angles up to a global gauge; in phase retrieval it concerns whether multi-plane measurements can be digitally registered to a single underlying complex field; in temporal video understanding it concerns whether action phases admit a monotone alignment path; and in matrix-phase control it concerns whether a common right multiplier can confine several matrices to a prescribed phase sector (Marron et al., 2015, Gao et al., 2019, Wang et al., 2024, Dave et al., 2024, Wu et al., 18 Jul 2025).

1. Scope of the concept

A useful cross-domain reading is that phase alignability concerns compatibility between observed phase structure and a restricted transformation class. The transformation may be a time-warp, a global phase, a local unitary, a projective registration, a rank-one phase history, or a common feedback matrix. What changes between fields is not the existence of phase as such, but the admissible action used to remove, estimate, or stabilize it.

Domain	Operational meaning	Representative papers
Functional and temporal data	Existence of “reasonable” monotone warps that reduce phase variability while preserving amplitude structure	(Marron et al., 2015, Dave et al., 2024, Dave et al., 2024)
Synchronization and phase recovery	Recovery of unknown angles or Fourier phases modulo a global gauge or circular shift	(Gao et al., 2019, Shahverdi et al., 2024)
Optical and quantum systems	Registration or compensation so that one field or Bell-state phase explains all observations	(Wang et al., 2024, Nai et al., 14 Apr 2026)
Matrix phase and control	Existence of a common $K$ such that matrix phases lie in a prescribed sector with rank preserved	(Wu et al., 18 Jul 2025, Mao et al., 25 Jul 2025)
Social orientation dynamics	Stability and switchability of side-by-side versus face-to-face phases	(Sarker et al., 2 Jun 2025)
InSAR phase linking	Degree to which all pairwise interferometric phases are explained by a single-reference phase history	(Heimpel et al., 28 Oct 2025)

This suggests that phase alignability is best treated as a structural property rather than a single metric. In some domains it is fundamentally an identifiability statement; in others it is a controllability statement; in others it is a goodness-of-fit statement for a constrained latent representation.

2. Temporal registration and amplitude–phase separation

In functional data analysis, phase variation is defined as lateral displacement of features such as peaks, valleys, or threshold crossings, while amplitude variation refers to vertical differences in height, peak magnitude, or shape. If observed functions $y_i$ contain both effects, amplitude-only functions $x_i$ satisfy $x_i(t)=y_i(h_i(t))$, where $h_i$ is a monotone time-warp that inverts observed clock-time to system-time. On $[0,1]$, the admissible warp set is

$$\Gamma=\{\gamma:[0,1]\to[0,1]\mid \gamma(0)=0,\ \gamma(1)=1,\ \gamma\text{ is a diffeomorphism},\ \gamma'(t)>0\}.$$

The central warning is that naïve alignment by minimizing $\|f-g\circ\gamma\|_{\mathbb L^2}^2$ is prone to pinching, asymmetry, and non-metric behavior. The alternative developed around Fisher–Rao geometry uses the SRSF/SRVF representation and an amplitude distance

$$d_A([q_1],[q_2])=\inf_{\gamma\in\Gamma}\ \bigl\|q_1-(q_2\circ\gamma)\sqrt{\gamma'}\bigr\|_{\mathbb L^2},$$

thereby separating reparameterization from amplitude in quotient space (Marron et al., 2015).

The same logic reappears in fine-grained action recognition, where two videos are temporally alignable if their phase sequences can be matched under a monotone time-warp. Standard DTW provides a pairwise alignment cost,

$$D_{\mathrm{DTW}}(X,Y)=\min_\pi \sum_{(i,j)\in\pi} c(x_i,y_j),$$

but the literature emphasizes that raw DTW is pairwise, path-length dependent, and not calibrated as a verification probability. FinePseudo therefore uses soft-DTW on a cosine-derived cost matrix and maps the resulting distance through a learned alignability score $y_i$0, which is then used for pseudo-label refinement in semi-supervised fine-grained action recognition (Dave et al., 2024).

Alignable Video Retrieval extends the notion from pair selection to search. Given a query video $y_i$1 and candidate $y_i$2, temporal alignability is defined by the existence of a low-cost monotone path through a frame-to-frame cost matrix. The paper operationalizes this through DRAQ,

$y_i$3

where $y_i$4 is the optimal DTW cost and $y_i$5 is the empirical mean cost of random monotone paths. Low DRAQ indicates that the optimal path is substantially better than a biased random baseline, which is presented as a more direct measure of alignability than semantic similarity alone (Dave et al., 2024).

3. Synchronization, phase recovery, and identifiability

In phase synchronization, alignability means recoverability of node phases from noisy pairwise measurements modulo a global phase. For $y_i$6, classical single-frequency formulations optimize $y_i$7 over unit-modulus vectors, whereas the multi-frequency formulation lifts the problem to harmonics,

$y_i$8

Under the unified sub-Gaussian model $y_i$9, alignability is tied to graph connectivity, the noise regime $x_i$0, and a gap-control condition guaranteeing sharp periodogram peaks. The main guarantee yields $x_i$1, showing that harmonics improve identifiability by sharpening the Dirichlet-kernel peak and enforcing cross-frequency consistency (Gao et al., 2019).

In multireference alignment, the corresponding issue is recovery of Fourier phase from noisy, randomly circularly shifted observations. The method of moments fixes the power spectrum but leaves phases undetermined; the cited work therefore interprets phase recovery as optimization on the power-spectrum manifold

$x_i$2

with iterative template alignment and projection. In Fourier space, projection preserves the current phase and replaces magnitudes by the estimated power spectrum, while cross-correlation supplies shift estimates. This makes phase alignability a question of whether the true phase configuration is recoverable on the resulting torus geometry, up to an unavoidable global circular shift (Shahverdi et al., 2024).

The literature also shows that alignability is not reducible to phase-only concentration measures. Warped phase coherence modifies the analytic signal by $x_i$3, thereby introducing controlled sensitivity to amplitude fluctuations and average relative phase. In coupled oscillator groups, collective alignability can invert the sign suggested by microscopic coupling: the reduced collective interaction takes the form $x_i$4, and anti-phase collective synchronization can occur in spite of microscopic in-phase external coupling, and vice versa (Minati et al., 2019, Kawamura, 2014). A different statistical realization appears in recurrence-based music analysis, where high correlation of normalized $x_i$5 sequences, specifically $x_i$6, is treated as operational evidence that the phases of two non-identical sitar performances are alignable despite strong differences in tempo, timbre, and dynamics (Mukherjee et al., 2014).

4. Optical, interferometric, and quantum formulations

In multi-plane phase retrieval, phase alignability is the ability to align intensity measurements across multiple planes in a manner consistent with a single underlying complex object field and its physics-based propagation. The cited ACC strategy addresses axial and lateral misalignment by refocusing each plane into object space, detecting SIFT features there, estimating neighboring-plane transforms, and composing them in a cascade $x_i$7. The key claim is that homography on raw defocused intensities is inadequate because diffraction is a wavefield operator, not a scene-independent 2D projective warp. Alignability is therefore increased by performing registration in refocused sample space before imposing amplitude constraints in the retrieval loop (Wang et al., 2024).

RIS-assisted communication formulates the same issue in discrete beamforming terms. The ideal continuous phase for RIS element $x_i$8 is

$x_i$9

whereas the discrete problem uses $x_i(t)=y_i(h_i(t))$0-bit phase shifts with quantization interval $x_i(t)=y_i(h_i(t))$1. The optimality law is explicit: for the optimal discrete solution,

$x_i(t)=y_i(h_i(t))$2

This leads to DTPQ, which searches the $x_i(t)=y_i(h_i(t))$3 candidate thresholds induced by the continuous phases themselves, and EIPQ, which searches a small equal-interval set of thresholds. The paper further states that the DoF of the optimal solution is proportional to the number of unit cells linearly and is irrelevant to the number of bits (Sang et al., 2023).

Diffractive interferometry and entanglement-based quantum communication introduce compensation rather than estimation. For diffraction gratings, a standard Gaussian modal decomposition reproduces the correct amplitude redistribution under small displacement but omits the geometric phase unless the measurement coordinates are moved with the effective beam displacement or the field is multiplied by the geometric factor. For lateral translation, the missing term is the grating-translation phase

$x_i(t)=y_i(h_i(t))$4

In Bell-state QKD, phase alignability means that local unitaries $x_i(t)=y_i(h_i(t))$5 remove an arbitrary relative phase and map a phase-shifted Bell state to a standard Bell state. The corresponding BBM92 error formula is

$x_i(t)=y_i(h_i(t))$6

and geometric-phase-assisted compensation experimentally restores QBER to approximately $x_i(t)=y_i(h_i(t))$7, below the $x_i(t)=y_i(h_i(t))$8 threshold required for secure QKD (Lodhia et al., 2013, Nai et al., 14 Apr 2026).

5. Matrix phase, control, and networked systems

A matrix-phase version of alignability has recently been formalized. For a set $x_i(t)=y_i(h_i(t))$9 with ranks $h_i$0, simultaneous $h_i$1-alignability means the existence of a single $h_i$2 such that

$h_i$3

for all $h_i$4. The associated diversity is

$h_i$5

and feasibility is checked through LMIs involving $h_i$6 and $h_i$7. This turns phase alignability into a constrained clustering criterion: agents or matrices are grouped when a common $h_i$8 can place them inside a small sector (Wu et al., 18 Jul 2025).

The same language drives heterogeneous multi-agent synchronization. Agents share linear common oscillatory modes, and the residue sets $h_i$9 at each mode are used to measure diversity. Interaction quality is quantified by the essential phase $[0,1]$0 of the graph Laplacian block. The main sufficient condition is the phase trade-off

$[0,1]$1

under which low-gain, component-wise synchronizing controllers exist. When this fails, the paper proposes a small controller inventory via clustering, rather than one controller per agent (Mao et al., 25 Jul 2025).

A closely related networked notion appears in coherent wireless superposition. There, phase alignability is the ability of multiple transmitters to pre-rotate their signals so that received phases align to a desired angle over a finite time window. Phase-coded pilots eliminate the round-trip phase offset, but residual CFO remains dominant: under the CFO-resilient schedule,

$[0,1]$2

This makes alignability explicitly time-limited, with a coherence window that contracts as residual CFO, mobility, or node count increases (Sahin, 27 Jun 2025).

6. Diagnostics, phase transitions, and recurrent limitations

Because the term spans multiple mathematical settings, its diagnostics are domain-specific. In FDA, suggested diagnostics include variance decomposition, amplitude distance $[0,1]$3, phase distance

$[0,1]$4

alignment energy $[0,1]$5, and the signal-to-warp ratio. In multi-temporal InSAR, alignability is assessed by whether all pairwise interferometric phases can be explained by a single-reference phase history; the proposed coefficients are the closure phase coefficient $[0,1]$6, the goodness-of-fit coefficient $[0,1]$7, and the ambiguity coefficient $[0,1]$8, all normalized to $[0,1]$9 with explicit noise-floor correction (Marron et al., 2015, Heimpel et al., 28 Oct 2025).

Socially driven motion shows that alignability can itself undergo a phase transition. The cited preschool-classroom data reveal a sharp threshold at approximately $$\Gamma=\{\gamma:[0,1]\to[0,1]\mid \gamma(0)=0,\ \gamma(1)=1,\ \gamma\text{ is a diffeomorphism},\ \gamma'(t)>0\}.$$0: below it, dyads predominantly align side-by-side, and above it, face-to-face orientations prevail. The inferred minimal pseudo-potential

$$\Gamma=\{\gamma:[0,1]\to[0,1]\mid \gamma(0)=0,\ \gamma(1)=1,\ \gamma\text{ is a diffeomorphism},\ \gamma'(t)>0\}.$$1

yields a bifurcation when $$\Gamma=\{\gamma:[0,1]\to[0,1]\mid \gamma(0)=0,\ \gamma(1)=1,\ \gamma\text{ is a diffeomorphism},\ \gamma'(t)>0\}.$$2 changes sign. In this setting, alignability means stability, depth, and switchability of the corresponding minima, rather than parameter estimation or compensation (Sarker et al., 2 Jun 2025).

Several recurrent limitations cut across the literature. First, amplitude–phase decompositions need not be unique without scientific priors; the same variability may be encoded as phase or amplitude (Marron et al., 2015). Second, overly flexible transformation classes overfit: free diffeomorphisms cause pinching in functional alignment, raw DTW forces alignments that are not class-discriminative, and homography on raw diffractive intensities ignores wave propagation (Marron et al., 2015, Dave et al., 2024, Wang et al., 2024). Third, many formulations are intrinsically gauge-ambiguous: synchronization is unique only up to a global phase, multireference alignment only up to a circular shift, and Bell-state compensation only up to locally irrelevant global factors (Gao et al., 2019, Shahverdi et al., 2024, Nai et al., 14 Apr 2026). This suggests that phase alignability is most informative when reported together with the admissible transformation group, the chosen objective, and diagnostics of stability or ambiguity.