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Streaming Low-Rank Matrix Approximation with an Application to Scientific Simulation

Joel A. Tropp, Alp Yurtsever, Madeleine Udell, Volkan Cevher

arXiv:1902.08651v1math.NAcs.NA

TL;DR

Large scientific data matrices can be costly to store and compress after collection. The paper presents a sketch-based streaming algorithm with error analysis and estimation; experiments report improvement over existing techniques and support its use on scientific simulation and measurement data.

  • Problem

    Scientific data matrices may be too large to store, process, or transmit, while conventional full-matrix compression incurs high storage and communication costs.

    Source: §1 · p. 1, §1.2.3 · p. 4

  • Method

    The paper develops a new sketch-based algorithm for low-rank approximation from streaming data, with a priori error bounds and an a posteriori estimator.

    Source: §8 · p. 29

  • Results

    The new method improves over existing techniques in numerical experiments.

    Source: §8 · p. 29

  • Takeaways & Limitations

    Experiments with a Navier–Stokes simulation and sea-surface-temperature data support on-the-fly compression of large-scale scientific simulations and data collection.

    Source: §8 · p. 29

    The initial approximation can contain spurious information in its smaller singular values and associated singular vectors.

    Source: §2.6 · p. 7

Abstract

from arXiv · show

This paper argues that randomized linear sketching is a natural tool for on-the-fly compression of data matrices that arise from large-scale scientific simulations and data collection. The technical contribution consists in a new algorithm for constructing an accurate low-rank approximation of a matrix from streaming data. This method is accompanied by an a priori analysis that allows the user to set algorithm parameters with confidence and an a posteriori error estimator that allows the user to validate the quality of the reconstructed matrix. In comparison to previous techniques, the new method achieves smaller relative approximation errors and is less sensitive to parameter choices. As concrete applications, the paper outlines how the algorithm can be used to compress a Navier--Stokes simulation and a sea surface temperature dataset.

1. Motivation.

Scientific data matrices can be too large to store or revisit, yet often admit useful low-rank approximations. The paper develops streaming sketching methods to construct and validate such approximations, with applications to simulation and measurement data.

1.2.3. Compression of Scientific Data: Current Art.

  • Scientific matrices can be too large to store, process, or transmit; collecting the full matrix before compression incurs high storage and communication costs.Snapshot and rank-one SVD updating methods may fail to preserve a complete, consistent view, and rank-one updating is computationally costly at each step.Source: §1 · p. 1, §1.2.3 · p. 4

1.1.1. Streaming.

  • The streaming model presents a matrix as simple linear updates that cannot be revisited, while simulation dimensions and time horizons can be large or unknown in advance.The updates are often sparse, low-rank, or otherwise structured.Source: §1.1.1 · p. 2, §1.2.1 · p. 3

1.1.2. Sketching.

  • A fixed randomized linear sketch compresses the matrix, and linearity lets each innovation be processed by adding its sketch to the current summary.Structured innovations can make this update efficient.Source: §1.1.2 · p. 2

1.1.5. Contributions.

  • The algorithm combines methods from [71, Thm. 12] and [69, Alg. 7], treats matrix dimensions symmetrically, and can extend to streaming Tucker approximation.The paper frames accurate truncated-SVD approximation with minimal resource use as important for scientific simulation and data analysis.Source: §1.1.5 · p. 3
  • The new method can achieve near-optimal relative approximation on matrices with decaying singular values, is especially accurate at small storage budgets, and is less sensitive to sketch parameters and truncation rank.A priori error bounds support parameter choices; the paper also reports higher-quality estimates of leading singular vectors.Source: §1.1.5 · p. 3

1.1.3. Matrix Approximation.

  • An a posteriori estimator validates approximation quality and provides a principled way to choose the final rank.The method also uses a target rank as a rough estimate when selecting sketch size.Source: §1.1.3 · p. 2, §1.1.5 · p. 3

1.2.2. On-the-Fly Compression via Sketching.

  • During a simulation, each rank-one innovation can update the sketch at negligible incremental cost; after the run, the sketch supports a low-rank approximation and an error estimate.The paper presents sketching as a way to maintain a summary during data generation and extract a near-optimal approximation afterward.Source: §1 · p. 1, §1.2.2 · p. 4

1.2.4. Contributions.

  • Applications include a 430 MB Navier–Stokes simulation, a 1.1 GB weather-station temperature dataset, a phase-retrieval problem, and a sea-surface-temperature matrix exceeding 75 GB.The sea-surface-temperature experiment required no parameter adjustment or retrospection.Source: §1.2.4 · p. 4

2. Sketching and Low-Rank Approximation of a Matrix.

The section develops a streaming sketch that captures the matrix’s range, co-range, and core, then reconstructs a truncated low-rank approximation. Its design uses independent core information and truncates after core estimation, with a permanence property for rank-r approximations.

2.1. Dimension Reduction Maps.

  • A randomized linear dimension-reduction map reduces vectors from dimension N to d while preserving Euclidean distances on average; the input requires a target rank r0 that influences sketch size.The framework assumes known matrix dimensions for simplicity but can be modified for growing dimensions.Source: §2.1 · p. 5, §2.2 · p. 5

2.3. The Sketch.

  • An independent core sketch Z improves estimates of A’s singular values and vectors, while X and Y capture its co-range and range.The method forms the core estimate from Z and truncates the initial approximation after estimating the core.Source: §2.3 · p. 6, §2.8.1 · p. 9

2.4. Linear Updates.

  • For a streaming matrix, linear updates can update the sketch directly; the update cost depends on exploiting structure such as sparsity or low rank in each innovation.A fixed linear transform of the stream can also be tracked by sketching its matrix image.Source: §2.4 · p. 6

2.5. Optional Step: Centering.

  • Centering can remove trends such as temporal averages, and the paper recommends doing so before sketching for superior accuracy.For row-wise centering, the stream maintains each row’s mean and updates the sketches with the centered innovations.Source: §2.5 · pp. 6–7

2.6. Computing Truncated Low-Rank Approximations.

  • The core estimate C is obtained from Z using least-squares computations, then combined with orthonormal range and co-range bases to form the initial rank-k approximation.The method estimates the core because P and Q depend on A and Q*AP cannot be computed directly from a linear sketch.Source: §2.6 · p. 7, §2.7 · p. 8
  • The initial rank-k approximation can contain spurious information in its smaller singular values and associated vectors, so the method takes its best rank-r truncation for a more reliable approximation.The final truncation is formed by replacing the initial approximation with its best rank-r approximation in Frobenius norm.Source: §2.6 · p. 7
  • The rank-r truncation persists inside every higher-rank approximation, unlike earlier reconstruction methods whose rank-r results can vary wildly with r.The paper reports that its new truncated approximation matches or improves on earlier techniques in numerical experiments.Source: §2.6 · pp. 7–8

2.8. Discussion of Related Work.

  • Compared with earlier NLA methods, the algorithm uses an independent core sketch and truncates after core estimation; sketch-and-solve methods are often unsuitable for high-accuracy computations.The authors also identify parameter selection and numerically sound implementation as practical challenges addressed in the paper.Source: §2.8.1 · p. 9, §2.8.2 · p. 9

3. Randomized Linear Dimension Reduction Maps.

The section describes Gaussian, scrambled SRFT, and sparse sign dimension-reduction maps for sketching. Their trade-offs include map storage and application costs, analysis completeness, data movement, and implementation requirements.

3.1. Gaussian Maps.

  • Gaussian maps are simple, effective, and fully analyzable, but their storage and arithmetic costs make them less appealing when output dimension d is large.Applying a map to a vector costs O(dN) arithmetic operations.Source: §3.1 · p. 10
  • Gaussian and sparse maps can be extended to vectors with previously unknown dimensions because their columns are statistically independent.In a stream, previously unseen coordinates can require enlarging the map’s domain.Source: §3.1 · p. 10

3.2. Scrambled SRFT Maps.

  • SSRFT maps are recommended for practical implementations; they store O(N) numbers and apply in O(N log N) operations, but their analysis is less complete than Gaussian maps.Fast trigonometric transforms and permutation-driven data movement can challenge distributed use.Source: §3.2 · p. 10

3.3. Sparse Sign Matrices.

  • Sparse sign matrices are recommended when data movement is a concern, and their independently generated columns support reduced data movement and blocked matrix multiplication.They require sparse data structures and arithmetic to realize this benefit.Source: §3.3 · p. 11

4. Implementation and Costs.

The section describes the sketch’s storage and update costs, then outlines the computations and costs for initial and truncated approximations. Update expense depends on the structure of each incoming innovation.

4.1. Sketching and Updates.

  • The sketch stores four dimension-reduction maps and three matrices, with dense sketch matrices requiring k(m+n)+s^2 floating-point numbers.SSRFTs or sparse sign matrices are recommended to reduce storage for the maps.Source: §4.1 · p. 11
  • Updating the sketch is dominated by computing ΦH and HΨ, with the precise cost depending on the innovation’s structure and the dimension-reduction map.Innovations are often low-rank, sparse, or structured.Source: §4.1 · p. 11

4.2. The Initial Approximation.

  • Computing the initial rank-k approximation adds storage for P and Q and a core matrix C, and QR factorizations usually dominate arithmetic cost.When s is much larger than k, forming C can instead dominate at O(ks^2) operations.Source: §4.2 · pp. 11–12

4.3. The Truncated Approximation.

  • The truncated approximation takes r as input and returns a truncated SVD; its working storage and typical arithmetic cost are dominated by computing the initial approximation.For s much larger than k, a randomized SVD may be needed to achieve the stated arithmetic cost.Source: §4.3 · p. 12

5. A Priori Error Bounds.

The section develops error bounds and parameter guidance for reliable low-rank reconstruction from sketches. It treats initial and truncated approximations, then allocates a fixed storage budget for general and flat spectra.

5.1. Notation.

  • The analysis uses parameters for real and complex fields in a common formula, with singular-value tail energy providing the spectrum notation.The error bound's minimum over ϱ captures spectral decay as k increases, an effect described as more significant than reducing the first constant by adjusting s.Source: §5.1 · p. 13, §5.2 · p. 14

5.2. Analysis of Initial Approximation.

  • Under Gaussian dimension-reduction maps and s ≥ 2k + α, Theorem 5.1 bounds the expected error of the initial rank-k approximation for any input matrix.The analysis provides explicit constants for selecting sketch sizes; the section also notes that large deviations above the expected-error bound have negligible probability.Source: §5.2 · pp. 13–14
  • With k = 4r0 + α and s = 2k + α, the expected rank-k approximation error is within a factor 10/3 of the optimal rank-r0 approximation.For rapidly decaying spectra this parameter choice can be effective in practice; increasing k/r0 and s/k can drive the leading constants in the bound toward one.Source: §5.2 · p. 14

5.3. Analysis of Truncated Approximation.

  • The rank-r truncation inherits similar accuracy from an accurate initial approximation and can have very small relative error when the input spectrum decays.The result is presented as a bound for the truncated approximation under Theorem 5.1's assumptions.Source: §5.3 · pp. 14–15

5.4.2. General Spectrum.

  • For a fixed storage budget, the sketch matrices' dimensions determine the available budget, while structured dimension-reduction maps have a storage cost independent of k and s.For a general spectrum, the section maximizes k subject to s ≥ 2k + α and k(m + n) + s2 = T; the resulting parameter choice is described as suitable for a wide range of examples.Source: §5.4.1 · p. 15, §5.4.2 · p. 15

5.4.3. Flat Spectrum.

  • When the spectrum is known to flatten beyond an index ϱ̂, the section selects parameters by numerically solving a storage-constrained optimization problem tailored to that flat-spectrum point.The optimization imposes s ≥ 2k + α and k ≥ ϱ̂ + α + 1 while meeting the fixed budget T.Source: §5.4.3 · p. 15

6. A Posteriori Error Estimation.

The section adds an independent sketch to estimate reconstruction error after streaming, then uses that estimate to assess accuracy and select a truncation rank. It analyzes estimator reliability and uses the initial approximation to diagnose spectral decay.

6. A Posteriori Error Estimation.

  • A small independent Gaussian sketch is used to estimate the Frobenius norm of the approximation error, following Martinsson's proposed approach.The paper's analysis of the Gaussian distribution refines the analysis of estimators computed from this sketch.Source: §6 · p. 16

6.1. The Error Sketch.

  • An independent Gaussian error sketch supports a simple Frobenius-norm error estimate that can help choose the final truncation rank and validate the reconstruction.The error sketch is maintained through linear updates; its test matrix and sketch require q(m + n) floating-point numbers.Source: §6 · p. 16, §6.1 · p. 16

6.2. A Randomized Error Estimator.

  • The randomized estimator can be computed efficiently for a factored rank-r approximation, typically in O(qr(m + n)) arithmetic operations.The approximation must not depend on the error-sketch matrices Θ and W.Source: §6.2 · p. 16

6.4. The Error Estimator, in Probability.

  • The error estimator provides probabilistic information about squared Schatten 2-norm approximation error, with bounds on extreme underestimates and overestimates.For example, the estimate is below 0.1× the true error with probability less than 2^-βq, and exceeds 4× the true error with probability less than 2^-βq.Source: §6.2 · p. 17, §6.4 · p. 17

6.3. The Error Estimator: Mean and Variance.

  • The paper gives bounds on estimator mean and variance and on extreme values, supporting constant-size error sketches and confidence bands.The stated rationale for the probability bounds is rotational invariance of the Schatten norms and the standard normal distribution.Source: §6.3 · p. 17, §6.4 · p. 17

6.5. Diagnosing Spectral Decay.

  • The estimator tracks the actual scree curve when r ≪ k and reliably overestimates it by a modest amount.The initial approximation supplies a proxy for the input's tail energy, while the error estimator supports a more rigorous upper estimate and principled rank selection.Source: §6.5 · pp. 17–18

7. Numerical Experiments.

The experiments compare sketching methods across synthetic and application matrices, assessing reconstruction accuracy, parameter robustness, rank truncation, and a posteriori error estimates.

7.1. Alternative Sketching and Reconstruction Methods.

  • The new method uses three sketches with parameters k and s, storage T = k(m + n) + s^2, and extends [Upa16] and [TYUC17].The compared methods have similar arithmetic costs, so the experiments focus on storage rather than runtime.Source: §7.1 · pp. 18–19

7.2. Experimental Setup.

  • Experiments cover synthetic matrices with varied spectral decay and application matrices including Navier–Stokes flow, weather, sea-surface temperature, and optimization outputs.The synthetic trials average 20 runs; oracle error is the best relative error at a fixed storage budget and is not attainable in practice.Source: §7.3.1 · pp. 19–20, §7.2.1 · p. 19, §7.2.2 · p. 19

7.4. Insensitivity to Dimension Reduction Map.

  • At oracle parameter values, reconstruction is insensitive to the dimension-reduction map, and theory-based sketch parameters almost attain oracle error.These findings support transferring Gaussian-map results to SSRFT and sparse maps, and using theoretically selected parameters in applications.Source: §7.4 · p. 21, §7.5 · p. 21

7.6. Comparison of Reconstruction Formulas: Synthetic Examples.

  • Across most synthetic inputs, the proposed method improves over earlier methods, sometimes by orders of magnitude; gains are strongest for faster spectral decay and small storage budgets.For slow-decay cases it matches [Upa16] and can be slightly worse than [HMT11]; it also performs better than [TYUC17] at small storage budgets.Source: §7.6 · p. 21, Figure 1 · p. 22

7.7. Comparison of Reconstruction Formulas:

  • On real examples, the proposed method improves over competitors with both oracle and theoretical parameter choices, with especially striking gains for effectively rank-deficient matrices.Its benefits are least pronounced for MinTemp; the comparison includes oracle-error curves and natural-parameter curves, except for [HMT11].Source: §7.7 · pp. 22, 25, Figure 2 · p. 23

7.8. Example:

  • For StreamVel, the sketch yields a rank-10 approximation with relative Frobenius error below 9.2 · 10^-3 using 5.8 MB rather than 409.7 MB, a 70.6× compression rate.The leading singular vectors from the sketch closely match those from the full matrix, with small discrepancies among higher vectors.Source: §7.8 · p. 25

7.9. Rank Truncation and A Posteriori Error Estimation.

  • For StreamVel, truncating to r ≤ k/4 achieves relative error below 10^-2 across a range of k; the untruncated rank-k approximation can worsen as k increases.The rank-k approximation underestimates tail energy near r ≈ k, while empirical scree bounds track the actual curve and help select truncation rank.Source: §7.7 · p. 25, §7.9 · pp. 26, 28

7.10. Example: Sea Surface Temperature Data.

  • Error estimators are unbiased but highly variable; doubling error-sketch size q halves estimate spread, while approximation errors cluster tightly, and truncated rank-r approximations are more successful than initial rank-k approximations.Source: §7.9 · p. 29

1. Parameter selection.

The SeaSurfaceTemp experiment uses a storage budget T = 48(m + n), with natural parameters k = 47 and s = 839 and sparse dimension-reduction maps.

2. Parameter selection.

  • The SeaSurfaceTemp experiment uses T = 48(m + n), natural parameters k = 47 and s = 839, sparse maps, and an error sketch of size q = 10.These are the selected storage and sketch settings for the sea-surface temperature example.Source: §1 · p. 29

2. Data collection.

The data are streamed one year at a time to build the approximation and error sketches.

3. Data collection.

  • The data are streamed one year at a time to construct both the approximation and error sketches.This is the collection procedure used for SeaSurfaceTemp.Source: §2 · p. 29

3. Error estimates and rank truncation.

The rank-5 truncation is selected by visually inspecting empirical scree bounds, which are expected to bracket the unknown curve.

3. Error estimates and rank truncation.

  • Visual inspection of empirical scree estimates sets truncation rank r = 5, expected to capture all but 0.04% to 0.8% of the energy.The lower and upper estimates should bracket the unknown scree curve while mimicking its shape.Source: §3 · p. 29

4. Visualization.

The SeaSurfaceTemp approximation’s singular vectors reveal interpretable temperature patterns, while the sketch compresses the matrix by 222×.

4. Visualization.

  • The first left singular vector represents the mean temperature profile, while the first right singular vector shows a warming trend.The second pair reflects the austral/boreal divide, and later vectors capture long-term climatological features.Source: §4 · p. 29
  • 4.09 · 10^7 numbers represent the approximation and error sketches, versus 9.09 · 10^9 in the matrix, a 222× compression ratio.The paper also reports modest computation time because the algorithm operates on substantially smaller matrices.Source: §4 · p. 29

8. Conclusions.

The paper presents a streaming sketch and reconstruction algorithm with error guarantees and reports experiments supporting its use for scientific data. The appendix develops supporting random-matrix facts and error analyses.

8. Conclusions.

  • The algorithm provides a priori error bounds for parameter selection and an a posteriori estimator for validating performance and selecting approximation rank.Experiments are reported to show improvement over existing techniques; Navier–Stokes and sea surface temperature examples support on-the-fly compression of scientific data.Source: §8 · p. 29

A.1. Facts about Random Matrices.

  • The appendix invokes a Gaussian random-matrix expectation identity from [35, Prop. A.1 and A.6] and [69, Fact A.1], under t > q + α.It specifies α = 1 over the reals and α = 0 over the complex numbers.Source: §A.1 · p. 31

A.2. Results from Randomized Linear Algebra.

  • The randomized low-rank approximation analysis uses Gaussian test matrices and establishes results for the range and co-range bases computed by (2.7).The argument draws on [35, Thm. 10.5] and [70, Sec. 8.2]; the appendix then sets up analysis of the core approximation error.Source: §A.2 · p. 31

A.3. Decomposition of the Core Matrix Approximation Error.

  • The core approximation error is decomposed under full-column-rank conditions on Φ1 and Ψ1, then analyzed probabilistically under distributional assumptions on Φ and Ψ.The derivation isolates the error by subtracting Q∗AP from the core sketch expression.Source: §A.3 · pp. 31–32

A.4. Probabilistic Analysis of the Core Matrix.

This section examines sketch-based low-rank reconstruction and its analysis, covering probabilistic bounds for core-matrix and compression errors, Gaussian-sketch error estimation, and implementation and comparison methods. It also describes the [TYUC17] sketch and rank-r reconstruction, noting its storage use and sensitivity to parameter sizes.

A.4. Probabilistic Analysis of the Core Matrix.

  • With independent standard-normal dimension-reduction matrices and s ≥ k, the core-matrix error C − Q*AP has expectation zero.The proof attributes this mean-zero result to the independence and zero mean of Φ2 and Ψ2.Source: §A.4 · pp. 32–33

A.5. Probabilistic Analysis of the Compression Error.

  • For ϱ < k − α, the expected compression-error bound combines the tail-energy term with two expected sketch-dependent terms.The bound includes ∥Σ2∥2^2, E∥Υ1†Υ2Σ2∥2^2, and E∥Σ2Ω2*(Ω1*)†∥2^2.Source: §A.5 · pp. 33–34

A.6. The Endgame.

  • Under s ≥ 2k + α and ϱ < k − α, the proof completes Theorem 5.1 by combining the core-matrix and compression-error bounds and optimizing over ϱ.The proof drops a nonpositive term under the stated sketch-dimension condition.Source: §A.6 · pp. 34–35

B.1. The Frobenius Norm Estimator.

  • A Gaussian sketch of the discrepancy A − Âout is developed as a randomized estimator of its Schatten 2-norm, with analysis of its mean, variance, and upper and lower tails.For matrices, the Schatten 2-norm is the Frobenius norm.Source: §B.1 · p. 35, §B.1.2 · p. 35, §B.1.3 · p. 36

B.2. Diagnosing Spectral Decay.

  • The square root of the tail energy is established as a Lipschitz function of the input matrix.Source: §B.2 · p. 37

Appendix C. Code & Pseudocode.

  • The appendices provide Matlab pseudocode for Gaussian, SSRFT, and sparse dimension-reduction maps, and note that electronic materials reproduce supplemental numerical results.Source: Appendix C · p. 37, Appendix D · p. 37

D.1.1. The [HMT11] Method.

  • The [HMT11] approach is not competitive with more modern techniques, partly because it truncates singular vectors to rank r at the first step.The discussion points to Figures 29 and 30 for this deficiency.Source: §D.1.1 · p. 39

D.1.2. The [TYUC17] Method.

  • The prior sketching method [69, Alg. 7] is algebraically, but not numerically, equivalent to Clarkson & Woodruff [20, Thm. 4.9], whose paper lacks reliable implementation instructions.The earlier work included theoretical analysis, parameter-selection prescriptions, and numerical evaluation.Source: §D.1.2 · p. 39
  • The [TYUC17] method forms sketches X = ΥA and Y = AΩ∗ using Υ ∈ Fℓ×m and Ω ∈ Fk×n, with k ≤ ℓ.Source: §D.1.2 · p. 39
  • It computes a rank-r approximation from the sketch using a thin orthogonal–triangular decomposition.The approximation is defined in equation (D.1).Source: §D.1.2 · p. 39
  • The method uses more storage than necessary because ℓ must be somewhat larger than k, and it can be sensitive to the relative sizes of k and ℓ.Source: §D.1.2 · p. 40

D.1.3. The [Upa16] Method.

The appendix describes [Upa16]’s sketch-and-solve approach and experiments comparing reconstruction methods, parameter choices, and dimension-reduction maps. Flow-field tests show that rank reduction affects methods differently.

D.1.3. The [Upa16] Method.

  • [Upa16] builds on a two-parameter sketch and uses a sketch-and-solve reconstruction, but it cannot achieve high relative accuracy even with rapid spectral decay.Reducing r can make its approximation less reliable.Source: §D.1.3 · p. 40

D.3. Insensitivity to the Dimension Reduction Map.

  • The proposed method’s oracle errors are nearly identical across Gaussian, SSRFT, and sparse maps, except SSRFT performs slightly better at very large storage budgets.The other reconstruction methods are also reported as insensitive to the dimension-reduction map.Source: §D.3 · p. 40

D.2. Spectra of Input Matrices.

  • Figure 9 plots singular-value spectra for synthetic classes and real-data classes, including MinTemp, StreamVel, MaxCut, and PhaseRetrieval.The synthetic classes include LowRank, PolyDecay, and ExpDecay with effective rank R = 10.Source: §D.2 · p. 40, Figure 9 · p. 41

D.4. Achieving the Oracle Performance.

  • For most synthetic examples, general theoretical parameters track oracle error closely; flat-spectrum parameters work somewhat better with slowly decaying tails.The theoretical choices are not entirely reliable at very small budgets and fall short of oracle parameters for examples with substantial tail energy.Source: §D.4 · p. 41

D.5. Algorithm Comparisons for Synthetic Instances.

  • The synthetic comparison evaluates four reconstruction formulas at oracle parameters across effective ranks R = 5 and R = 20, using Schatten 2- and ∞-norm relative errors.The cited text identifies the plotted cases but does not state their comparative outcomes.Source: §D.4 · p. 41, §D.5 · p. 42

D.6. Algorithm Comparisons for Real Data Instances.

  • The real-data comparison tests four reconstruction formulas at oracle and theoretically chosen parameters, with method-specific parameter selections described.The [Upa16] algorithm uses the same sketch with the proposed method’s parameter choice, while [HMT11] has no free parameters.Source: §D.6 · p. 42

D.7. Flow-Field Reconstruction.

  • At r = 5 and r = 10, (2.10) and [TYUC17] retain the same singular-vector estimates, while [HMT11] and [Upa16] worsen at r = 5.At r = 10, all methods give reasonable results, but [HMT11], [Upa16], and [TYUC17] estimates for ranks 6–9 begin to deviate from the original matrix.Source: §D.7 · p. 42

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