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Dynamics-Aware Sampling and Recovery for Tensor-Valued Dynamical Systems

Seok-Young Chung, Longxiu Huang, Yisen Wang

arXiv:2610.34621v1math.NAcs.ITcs.NAmath.IT

TL;DR

Recovering tensor-valued dynamical states from partial spatial observations requires accounting for shared eigenvalues across Fourier slices, which couple their recovery geometry. The paper develops deterministic and randomized recovery analyses around aggregated eigenspace blocks and evaluates Tensor DS on synthetic and real data. Experiments support the predicted sampling behavior and show that learned t-product dynamics can improve real-data reconstruction, subject to sampling and conditioning limits.

  • Problem

    The paper asks when partial spatial observations over time uniquely determine an evolving tensor’s initial state while preserving its tensor structure.

    Source: §1 · p. 2

  • Method

    The framework groups shared Fourier-slice eigenvalues into aggregated blocks and analyzes recovery through stacked-system factorization, blockwise conditions, and learned t-product dynamics.

    Source: §3.4 · p. 17, §7 · p. 38

  • Results

    Experiments on SST and turbulence show learned t-product dynamics can improve reconstruction, with relative advantage varying by sampling level and dataset smoothness.

    Source: §6.5 · p. 38, §7 · p. 38

  • Takeaways & Limitations

    Tensor dynamics are most effective when sampling yields an informative recovery system, while tuned graph-spectral baselines can remain strong on very smooth data.

    Source: §7 · p. 38

    The decoupled formulation requires T ≥ |Λ| and reliable spectral decomposition and grouping, and its inverse Vandermonde step may remain poorly conditioned.

    Source: §3.4 · p. 18

Abstract

from arXiv · show

We study sampling and recovery for tensor-valued dynamical systems evolving under the t-product, using partial spatial observations over time. Although the t-product Fourier transform yields slicewise matrix dynamics, recovery does not generally decouple across slices. Eigenvalues shared across Fourier-domain frontal slices couple the corresponding eigenspaces into aggregated spectral blocks, which determine the tensor-specific recovery geometry and space--time tradeoff. We establish deterministic identifiability conditions for the initial tensor state. In the diagonalizable case, these are frame conditions on the aggregated blocks. We extend them to non-diagonalizable systems through generalized eigenspaces and Jordan structure. A temporal--spectral factorization of the finite-horizon stacked recovery system underlying Tensor Dynamical Sampling (DS) yields a decoupled two-stage formulation once the temporal threshold is reached. Under independent spatial subsampling, we prove high-probability recovery guarantees using empirical Gram matrices on the aggregated blocks. The bounds depend on aggregated multiplicities, incoherence, and population conditioning. Stability estimates separate the spatial frame factor from the temporal Vandermonde factor and show that increasing temporal depth may worsen numerical conditioning even after identifiability is ensured. We implement Tensor DS and evaluate it on synthetic and real data. Phase transitions track the predicted blockwise sampling thresholds, and stability experiments illustrate Vandermonde conditioning. Experiments on sea surface temperature and turbulence velocity data show that a learned t-product system operator generally yields more accurate and structurally faithful reconstructions than vectorized low-rank, Tucker-type, graph-regularized, graph-spectral, and vectorized dynamical sampling baselines.

1 Introduction

The introduction frames recovery of tensor-valued evolving fields from partial spatial observations and explains why shared eigenvalues across Fourier slices create tensor-specific recovery blocks. It previews deterministic, randomized, stability, and computational analyses of that recovery problem.

1 Introduction

  • Tensor dynamical sampling asks whether partial spatial observations over time determine an initial tensor state, without flattening away the structure of evolving multidimensional fields.The paper considers t-product evolution observed on a spatial sampling set over a finite time horizon.Source: §1 · p. 2
  • Shared eigenvalues across Fourier-domain frontal slices have the same temporal signature, so their eigenspaces must be recovered jointly as aggregated spectral blocks.The dimensions of these blocks determine the spatial degrees of freedom the sampling pattern must capture.Source: §1 · pp. 2–3
  • The analysis combines fixed-mask frame conditions after temporal separation, independent Bernoulli spatial sampling guarantees, and stability estimates for temporal Vandermonde separation.The randomized bounds depend on aggregated spectral dimensions, eigenspace incoherence, and Gram-matrix geometry; temporal depth can still leave the Vandermonde inverse poorly conditioned.Source: §1 · p. 3

1.1 Related work

  • The paper extends preliminary stacked tensor sampling work [49] with deterministic identifiability, Jordan-structure, randomized-sampling, stability, and expanded experimental results.Related work also situates the study among classical dynamical sampling, multidimensional and graph-based sampling, and tensor-valued dynamical systems.Source: §1.1 · pp. 3–4

1.2 Contributions

  • The contributions include temporal-spectral factorization, necessary and sufficient frame conditions for aggregated blocks, Jordan extensions, Gram-matrix sampling guarantees, and stability analysis.The paper also implements Tensor DS with learned t-product operators and evaluates it on synthetic, sea surface temperature, and turbulence velocity data.Source: §1.2 · p. 4

1.3 Organization.

  • The paper proceeds from t-product and stacked-system preliminaries to deterministic theory, randomized sampling, stability, and synthetic and real-data experiments.Proofs and additional Jordan-structure and basis-conditioning discussions appear in the appendices.Source: §1.3 · pp. 4–5

2 Preliminaries and notation

The preliminaries set up the t-product system in the Fourier domain and show why shared eigenvalues, rather than individual frontal slices, organize recovery. A lateral-slice-wise stacked formulation provides the computational starting point for the paper’s temporal–spectral analysis.

2 Preliminaries and notation

  • The preliminaries introduce the tensor-algebra notation and summarize the principal symbols used throughout the paper.The t-product framework is recalled for third-order tensors.Source: §2 · p. 5, §2.1 · p. 5, Table 1 · p. 6

2.1 t-product and Fourier-domain dynamics

  • The third-mode DFT turns t-product evolution into slicewise matrix multiplication, providing the Fourier-domain representation used for the dynamics.The paper uses the standard unnormalized DFT convention consistent with MATLAB’s fft.Source: §2.1 · pp. 5–6

2.2 Spectral structure across frontal slices in Fourier domain

  • Shared eigenvalues across frontal slices define aggregated direct-sum eigenspaces, so independent recovery subproblems are indexed by eigenvalue rather than slice.After temporal samples are decoupled, the remaining spatial information for each eigenvalue lies on its aggregated block.Source: §2.2 · p. 8

2.3 The lateral-slice-wise stacked recovery system

  • For each lateral slice, recovery becomes a stacked system whose hidden temporal–spectral decomposition supports refined identifiability and stability analysis.The formulation decomposes the computation into p independent systems, one per lateral slice.Source: §2.3 · p. 9

3 Deterministic identifiability: necessary and sufficient conditions

This section develops deterministic identifiability tests for tensor dynamical recovery, relating finite-horizon observability to temporal–spectral factorization and spatial frame conditions on aggregated eigenspaces. It also contrasts direct stacked recovery with decoupled spectral reconstruction, highlighting how shared eigenvalues couple Fourier slices and affect spatial sampling requirements.

3.1 Lifted-matrix viewpoints: observability and classical dynamical sampling

  • Lifted observability and classical projected-frame tests give valid global recovery criteria, but do not reveal how Fourier-slice eigenspaces assemble into tensor-specific aggregated blocks.The tensor dynamics remain coupled across modes, motivating a Fourier-coordinate formulation with slice-aware sensing vectors.Source: §3.1 · pp. 10–11

3.2 The temporal–spectral structure of the stacked system

  • The stacked-system criterion separates spatial rank failure in aggregated λ-blocks from temporal ambiguity in the Vandermonde factor; at T ≥ |Λ|, only blockwise spatial rank remains.For T < |Λ|, recovery depends on whether the spatial range intersects the temporal factor’s nontrivial kernel.Source: §3.2 · p. 12
  • Unique recovery requires at least one spatial sample in every lateral slice, proving Conjecture 1 of [49].For the special lattice mask Ω = I × J × [n], this implies J = [p], yielding [49, Theorem 3.1].Source: §3.2 · p. 13

3.3 Temporal decoupling and frame conditions on aggregated eigenspaces

  • Once T ≥ |Λ|, unique recovery is equivalent to spatial frame conditions on each aggregated eigenspace block, whose dimension dλ grows when slices share eigenvalues.The temporal threshold is the number of distinct eigenvalues across all frontal slices; each lateral slice must cover the largest aggregated block.Source: §3.3 · p. 14
  • The frame condition depends on sample placement across both mode-1 directions and mode-3 Fourier phases, not only on the number of sampled entries.Mode-1 indices select eigenspace components within slice blocks, while mode-3 indices determine relative Fourier phases among blocks sharing an eigenvalue.Source: §3.3 · pp. 15–16

3.4 Reconstruction viewpoints from the stacked and decoupled systems

  • The stacked reconstruction recovers each lateral slice’s Fourier coefficients, assembles them into a Fourier-domain tensor, and applies an inverse DFT; the real-data implementation follows this approach [49].In the noiseless full-rank case, the finite-horizon system gives exact recovery; experiments use regularized least squares.Source: §3.4 · p. 17
  • The stacked formulation is the practical choice for learned real-data dynamics because it needs only finite-horizon matrices and avoids explicit diagonalization or eigenvalue grouping.It remains useful when the available time horizon is short.Source: §3.4 · pp. 17–18
  • When T ≥ |Λ|, the decoupled formulation first separates temporal spectral coefficients with a Vandermonde solve, then tests spatial frame conditions on aggregated eigenspace blocks.This two-stage view underlies the randomized Gram-matrix analysis and temporal-conditioning experiment.Source: §3.4 · pp. 17–18
  • Even when T ≥ |Λ|, decoupled recovery requires reliable spectral decomposition and eigenvalue grouping, and its inverse Vandermonde step may be poorly conditioned.The decoupled formulation explains the spectral mechanism, while the stacked formulation remains the real-data reconstruction framework.Source: §3.4 · p. 18

4 Random sampling robustness via Gram matrix concentration

Independent spatial subsampling yields high-probability recovery guarantees through empirical Gram matrices on aggregated eigenspace blocks. The sampling burden depends on block dimension, spatial incoherence, and population conditioning, rather than only on ambient dimension.

4 Random sampling robustness via Gram matrix concentration

  • The sampling model selects each sensor location independently with probability α and keeps the resulting spatial mask fixed throughout the observation horizon.This provides a simple alternative to combinatorial sensor-location design.Source: §4.1 · p. 18

4.1 Bernoulli sampling and population geometry

  • The population Gram matrix is block diagonal and positive definite, while its minimum conditioning and the size of each random summand govern the concentration guarantee.The incoherence parameter measures spatial spreading; small µλ corresponds to more evenly distributed sensing energy.Source: §4.1 · p. 19, §4.2 · p. 19

4.2 Incoherence and Matrix Chernoff bound

  • With T≥|Λ|, a Bernoulli sampling rate satisfying Theorem 4.2’s bound makes the empirical block Gram matrices nonsingular and ensures unique recovery with high probability.The bound depends on aggregated block dimension dλ, incoherence µλ, and population conditioning γλ,min.Source: §4.2 · p. 20
  • In the maximally incoherent Fourier-diagonalizable model, the sufficient sampling rate is governed by the largest aggregated block dimension dλ, not ambient dimension mn, and is sharp up to logarithmic factors.The phase-transition experiments closely track the theoretical blockwise sampling curve.Source: §4.2 · pp. 20–22
  • The analysis applies concentration after temporal decoupling to empirical Gram matrices on aggregated blocks, instead of using a global lifted-space coherence condition.This isolates spatial sensing geometry from temporal Vandermonde separation.Source: §4.2 · p. 22

5 Recovery stability and temporal conditioning

The stability analysis separates spatial frame quality from temporal Vandermonde conditioning. It shows that identifiability alone does not ensure stable recovery, and that temporal oversampling can worsen conditioning when spectral magnitudes are unbalanced.

5 Recovery stability and temporal conditioning

  • The reconstruction error has separate spatial and temporal factors: the spatial lower frame bound and the smallest singular value of the Vandermonde matrix.Thus, noise may be amplified by temporal separation even after T≥|Λ| ensures identifiability.Source: §5 · p. 23
  • Unequal temporal column norms can amplify noise during Vandermonde inversion, and spectral-magnitude imbalance alone can force conditioning to deteriorate as T grows.The estimate isolates this imbalance effect, not the full Vandermonde conditioning, which also depends on angular conditioning of normalized temporal profiles.Source: §5 · pp. 23–24
  • When ρ+ < 1, increasing T does not cause the analyzed Vandermonde-conditioning deterioration, although nearly colliding eigenvalues may still cause ill-conditioning.For distinct eigenvalues, the condition number is uniformly bounded for T≥|Λ| in this regime.Source: §5 · p. 24
  • In the mixed-modulus case 0≤ρ−<1<ρ+, the temporal depth needed to certify a conditioning threshold grows logarithmically in its inverse and increases as ρ+ approaches 1.The upper bound for the reciprocal condition number decays at least geometrically with T.Source: §5 · pp. 24–25

6 Numerical Experiments

Numerical experiments evaluate Tensor DS on synthetic data and real SST and turbulence velocity data, examining blockwise sampling trends, conditioning, and reconstruction fidelity.

6.1 Synthetic phase transitions and temporal stability

  • Across the synthetic phase diagrams, critical sampling decreases as m or n increases and increases with aggregated block dimension dλ, consistent with the predicted blockwise threshold.The diagrams use 40 trials per grid point and declare success below relative reconstruction error 10^-6; the experiments satisfy T ≥ |Λ| and use ρ+ < 1 to isolate spatial sampling effects.Source: §6.1 · pp. 25–27, Figure 2 · p. 26
  • After T reaches |Λ|, additional temporal samples can initially help two-stage recovery, but mixed-modulus eigenvalues can make Vandermonde conditioning worsen and reconstruction error rise as T grows.The experiment uses α = 0.7, a fixed spatial mask across time, and median errors over 20 noise realizations; T ≥ |Λ| ensures identifiability, not numerical stability.Source: §6.1 · p. 27, Figure 3 · p. 28

6.2 Real-data reconstruction setup

  • Real-data tests use monthly NOAA SST and JHTDB turbulence velocity data; Tensor DS learns a t-product operator by slice-wise truncated DMD, so these tests are algorithmic validation rather than direct tests of Theorem 4.2's exact assumptions.The SST setup uses 60 months for training and the next six for evaluation; turbulence frames are converted to velocity magnitude and a fixed 2D slice.Source: §6.2 · pp. 28–30

6.3 Sea surface temperature reconstruction

  • On SST, Tensor DS becomes one of the strongest performers in SNR, SSIM, and MAE at moderate-to-high sampling rates and maintains strong average accuracy across the six-month horizon.It is not uniformly dominant at the lowest reported sampling rate; the results describe strong performance once sampling is sufficient.Source: Figure 5 · p. 32, §6.3 · pp. 32–33, Figure 4 · p. 32
  • Tensor DS has the smallest monthly relative error in five of six SST months and is second-best in the remaining month at α = 0.6.SRS is often the closest competitor, while GR leads at Month 65; Tensor DS's advantage is therefore strong but not uniform across every month.Source: Table 3 · p. 33, §6.3 · p. 33
  • In SST visual comparisons, Tensor DS best preserves the large-scale warm band and smooth transitions, while competing methods show blurring, lost local contrast, or artifacts in the highlighted regions.SRS captures the dominant smooth component but loses some localized thermal contrast and warm-band boundary detail.Source: §6.3 · p. 33
  • On SST, Tensor DS preserves warm-band geometry and local ROI contrast, while SRS captures the smooth large-scale field but less sharply retains localized thermal contrast and boundary detail.POD-LD and Tucker lose some local thermal detail, while GR and Vector DS show visible artifacts in highlighted regions.Source: Figure 6 · p. 34

6.4 Turbulence velocity reconstruction

  • Tensor DS gives the strongest initial-state recovery across the velocity sampling range and remains the most stable and accurate method over the six-frame horizon.The comparisons report higher SNR and SSIM and lower MAE initially, with multi-step gains over low-rank, Tucker-type, and graph-based baselines.Source: §6.4 · pp. 35–36, Figure 7 · p. 36, Figure 8 · p. 36
  • At α = 0.6, Tensor DS yields the smallest framewise relative error at every step; Vector DS is near second-best early, while GR is typically second-best later.The velocity experiment uses magnitude frames from a two-dimensional slice of forced isotropic turbulence and evaluates a six-frame horizon.Source: §6.4 · pp. 34, 36, 37, Table 5 · p. 37

6.5 Summary of numerical findings

  • Synthetic experiments support the blockwise sampling scale governed by aggregated dimensions d_λ and show that identifiability and numerical stability differ after Vandermonde inversion.The real-data experiments extend the tensor dynamical modeling principle beyond the exact diagonalizable setting used in the theory.Source: §6.5 · p. 38
  • Across SST and velocity data, Tensor DS delivers stable, structurally faithful reconstructions, with its advantage varying by sampling level and dataset smoothness.SRS is competitive on smooth SST data but loses localized thermal contrast; Tensor DS separates more clearly from graph-spectral smoothing on textured velocity data.Source: Figure 6 · p. 34, Figure 9 · p. 37, §6.5 · p. 38

7 Conclusion

The conclusion identifies aggregated eigenspace blocks as the tensor-specific recovery geometry, shaping deterministic frame conditions, spatial sampling, and randomized recovery. Experiments support these results while highlighting a temporal-conditioning trade-off.

7 Conclusion

  • Shared eigenvalues across frontal slices create aggregated eigenspace blocks that determine the correct deterministic frame conditions and sampling burden.The blocks also govern the randomized recovery scale.Source: §7 · p. 38
  • Randomized sampling rates depend on aggregated-block dimensions, incoherence, and population conditioning, while greater temporal depth may worsen Vandermonde conditioning when spectral magnitudes are unbalanced.Temporal depth is needed for temporal separation, so it has a dual role.Source: §7 · p. 38
  • Synthetic phase transitions track predicted blockwise thresholds, and SST and turbulence experiments show learned t-product operators can improve practical reconstructions.The experiments also indicate tensor dynamics work best when sampling yields an informative recovery system; tuned graph-spectral baselines can be strong on very smooth data.Source: §7 · p. 38

8 Proofs of the main results

The proofs establish a temporal–spectral factorization of the stacked recovery matrix and use it to connect full rank with blockwise conditions. They also show that unique recovery requires at least one sampled coordinate for every spatial index.

8.1 Proof of Theorem 3.3

  • The stacked matrix factorization separates a temporal Vandermonde factor from a block-diagonal matrix of aggregated spectral blocks.The factorization is given after an invertible change of coordinates.Source: §8.1 · p. 39
  • Full column rank of the stacked matrix is equivalent to full column rank of each aggregated block together with the stated intersection condition.The proof transfers rank through the invertible coordinate change and proves both directions.Source: §8.1 · p. 40

8.2 Proof of Corollary 3.4

  • Unique recovery requires every spatial index j to have at least one sampled coordinate.If some index has no samples, its corresponding sensing block is zero, contradicting the rank condition.Source: §8.2 · p. 41

8.3 Proof of Proposition 3.5

The proofs use Fourier-domain slice dynamics and inverse Fourier expansion to express observed samples. For the diagonalizable recovery condition, temporal separation and a frame condition on concatenated eigenspaces yield sufficiency, while failure of the frame condition yields non-uniqueness.

8.3 Proof of Proposition 3.5

  • The t-product becomes slicewise multiplication in the Fourier domain, and inverse Fourier expansion gives the time-domain observation formula.The proof identifies this expression with the stated coefficients.Source: §8.3 · p. 41

8.4 Proof of Theorem 3.6

  • If the required frame condition fails, a nonzero tensor can produce the same sampled observations as the zero tensor, so unique recovery fails.This establishes necessity of the frame condition.Source: §8.4 · pp. 42–43
  • When T is at least the number of distinct eigenvalues, the Vandermonde matrix has full column rank, supporting the sufficiency argument.The proof then applies the frame condition to the concatenated eigenspace.Source: §8.4 · p. 42

8.5 Proof of Lemma 4.1

The randomized proof organizes reduced sensing vectors by eigenvalue blocks and applies matrix Chernoff bounds to empirical Gram matrices. Under diagonalizability and sufficient temporal depth, the resulting sampling condition gives high-probability recovery.

8.5 Proof of Lemma 4.1

  • For each eigenvalue, the reduced sensing vector has a block form, with indicator terms selecting matching indices.The lemma concludes by evaluating the resulting sum.Source: §8.5 · p. 43

8.6 Proof of Theorem 4.2

  • The empirical Gram matrix is analyzed under diagonalizability and T ≥ |Λ|, using independent Bernoulli spatial samples.The theorem states that its recovery condition holds with high probability when the sampling condition is met.Source: §8.6 · p. 44
  • The expected Gram sum has minimum eigenvalue αγλ,min, and the matrix Chernoff argument yields a sufficient sampling rate for recovery.The rate uses the incoherence relation Rλ ≤ μλdλ/(mn); the proof then combines conditions over spatial indices and eigenvalues.Source: §8.6 · pp. 44–45

8.7 Proof of Proposition 5.1

The stability proof separates temporal spectral separation from spatial frame recovery. Its bounds track Vandermonde conditioning as the temporal horizon grows and as spectral magnitudes differ.

8.7 Proof of Proposition 5.1

  • With T ≥ |Λ|, the Vandermonde matrix has full column rank, so least squares separates temporal spectral data before the λ-blocks are reconstructed spatially.The spatial frame lower bound then controls reconstruction of each block, yielding the desired error estimate.Source: §8.7 · pp. 45–46

8.8 Proof of Theorem 5.2

  • The proof relates the minimum and maximum norms of the temporal vectors cr to ϕT(ρ−) and ϕT(ρ+), respectively.This connects the spectral-magnitude extremes to the temporal conditioning factor ΦT.Source: §8.8 · p. 46
  • The stability factor ΦT decreases with T, and the proof shows its relevant bound tends to 0 when ρ+ > 1 and ρ−/ρ+ < 1.Here ρ− and ρ+ are the minimum and maximum eigenvalue magnitudes, respectively.Source: §8.8 · pp. 46–48

8.9 Proof of Corollary 5.3

The proof reduces the corollary to a bound from Theorem 5.2 and verifies when the stated condition holds across the temporal horizon.

8.9 Proof of Corollary 5.3

  • If condition (36) holds, then ΦT ≤ τ, so the conclusion follows from Theorem 5.2(i).The proof uses the notation ρ−, ρ+, ϕT, and ΦT from that theorem.Source: §8.9 · p. 48
  • In the remaining case, the proof establishes the equivalence used to verify condition (36).This completes the check of the condition needed for the theorem-based conclusion.Source: §8.9 · p. 48
  • When the stated spectral inequality holds, condition (36) follows automatically for every T ≥ R.The proof treats the remaining case by rewriting the condition.Source: §8.9 · p. 48

A Jordan structure across frontal slices in Fourier domain

For non-diagonalizable slice dynamics, recovery must account for generalized eigenspaces and nilpotent directions, not only eigenvalue amplitudes. The resulting identifiability criterion is a frame condition on aggregated generalized eigenspaces.

A Jordan structure across frontal slices in Fourier domain

  • With Jordan blocks, temporal samples include polynomial factors from the nilpotent structure, so recovery must include additional nilpotent directions alongside spectral amplitudes.The relevant spaces aggregate generalized eigenspaces associated with the same eigenvalue across slices.Source: A Jordan structure across frontal… · pp. 51–52
  • Unique recovery is characterized by frame conditions for each spatial index and eigenvalue on the corresponding aggregated generalized eigenspace.The theorem replaces the diagonalizable-case frame condition with one that captures the Jordan contributions.Source: A Jordan structure across frontal… · pp. 51–52
  • The extension supplies deterministic identifiability for non-diagonalizable dynamics, while the randomized theory is stated for the diagonalizable setting.The proof establishes sufficiency through triviality of the measurement null space and necessity by constructing a nonzero tensor with zero observations.Source: A Jordan structure across frontal… · pp. 52–54
  • Separating the Jordan contributions requires linear independence of the temporal sequences, equivalently invertibility of the associated confluent Vandermonde matrix.The sequences include powers of eigenvalues and polynomial factors indexed by the Jordan order.Source: A Jordan structure across frontal… · p. 53

B Conditioning across slices and basis scaling

The population conditioning parameter depends on how slice-wise eigenbases are scaled, so its interpretation requires a normalization convention. Unit-norm columns remove arbitrary scaling and make the resulting singular values invariant to residual phase or sign choices.

B Conditioning across slices and basis scaling

  • γλ,min is positive exactly when every slice-wise eigenbasis matrix Uλ,s has full column rank.The parameter is not purely a property of the eigenspace range because diagonal rescaling can change it.Source: B Conditioning across slices and basis… · pp. 54–55
  • Without fixed normalization, artificial diagonal rescaling can make γλ,min arbitrarily small and its sampling-rate lower bound arbitrarily pessimistic.Unit Euclidean norms for the eigenbasis columns remove this scaling freedom.Source: B Conditioning across slices and basis… · p. 55
  • Unit-norm eigenvectors give a diagonal rescaling whose condition number is within a factor √mλ,s of the best diagonal rescaling.After normalization, residual unit-modulus column scalings do not change singular values.Source: B Conditioning across slices and basis… · p. 55

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