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REVIEW 2 major objections 2 minor 76 references

PCA Gets Outlier Flags and Missing-Data Handling in One Rewrite

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2026-07-10 00:23 UTC pith:TPFQIMGB

load-bearing objection Useful integration of robust statistics into a practical PCA replacement; worth a serious referee to check the identifiability story. the 2 major comments →

arxiv 2607.08081 v1 pith:TPFQIMGB submitted 2026-07-09 astro-ph.IM astro-ph.SR

Robust Heteroskedastic Matrix Factorization: A Generalization of PCA that Flags Outliers and Handles Missing Data

Thomas Hilder , David W. Hogg , Andrew R. Casey , Hans-Walter Rix This is my paper
classification astro-ph.IM astro-ph.SR
keywords anomaliesdataoutliersrhmfrobustfactorizationflagsgeneralization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Robust Heteroskedastic Matrix Factorization (RHMF), a generalization of Principal Component Analysis (PCA) that simultaneously addresses three practical weaknesses of classical PCA: sensitivity to outliers, inability to handle per-feature measurement uncertainties or missing data, and lack of a built-in mechanism to identify which specific data points are anomalous. RHMF works through an iterative reweighting algorithm that implicitly maximizes a Student-t likelihood, which admits a hierarchical probabilistic interpretation where each data point carries its own latent variance. The method serves a dual purpose: it recovers a clean low-dimensional embedding of the data that is not corrupted by bad measurements, and it automatically flags both per-feature and per-object anomalies. The authors demonstrate the approach on synthetic data with different outlier classes and on real spectroscopic data from Gaia DR3, where it recovers known astrophysical anomalies such as Be-star binaries and M-dwarfs with subtle chromospheric emission in the Ca II triplet lines that would not be obvious to identify by eye.

Core claim

The central mechanism is the iterative reweighting algorithm that implicitly maximizes a Student-t likelihood. The Student-t distribution has heavier tails than the Gaussian assumed by standard PCA, which means it does not over-penalize data points that fall far from the model. This is equivalent to fitting a hierarchical model where each data point has its own latent variance: points that are well-explained by the low-dimensional structure get small latent variances (high weight), while genuine outliers get large latent variances (low weight). The reweighting thus serves double duty: it down-weights anomalous points so the recovered embedding stays clean, and the weights themselves become a

What carries the argument

The iterative reweighting loop that implicitly maximizes a Student-t likelihood, interpreted hierarchically as per-data-point latent variances that simultaneously clean the embedding and flag anomalies.

Load-bearing premise

The practical reliability of the anomaly-flagging capability depends on cross-validation successfully selecting good hyperparameters, and the paper demonstrates this on specific cases rather than proving it holds across all data regimes.

What would settle it

If cross-validation fails to reliably set hyperparameters on real-world datasets outside the specific demonstrations shown, the anomaly-flagging capability becomes unreliable.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Large astronomical surveys with heterogeneous data quality and inevitable bad pixels could use RHMF as a single preprocessing step that both cleans and flags, rather than separate outlier-rejection and dimensionality-reduction pipelines.
  • The per-feature anomaly flags could enable automated discovery of instrumental artifacts or calibration failures in survey data, since these manifest as systematic per-feature deviations rather than per-object anomalies.
  • Any domain using PCA on noisy or incomplete data, such as genomics, finance, or neuroscience, could adopt the same framework to get uncertainty-aware embeddings without switching to a fundamentally different methodological toolkit.
  • The hierarchical latent-variance interpretation connects the method to Bayesian robust regression, suggesting that prior distributions on the latent variances could encode domain-specific knowledge about expected outlier rates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Student-t implicit likelihood means RHMF has a continuous notion of outlierness rather than a binary in/out classification, which could be useful for ranking candidates in follow-up observation campaigns where telescope time is limited.
  • If the latent variances correlate with physically meaningful quantities, they could serve as discovery metrics in their own right, not just as nuisance parameters for cleaning the embedding.
  • The reliance on cross-validation for hyperparameter selection may become a bottleneck on very large datasets, motivating future work on approximate or streaming variants that update hyperparameters online.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Robust Heteroskedastic Matrix Factorization (RHMF), a generalization of PCA that is simultaneously robust to outliers, accommodates per-feature uncertainties and missing data, and automatically flags per-feature and per-object anomalies. The method uses iterative reweighting that implicitly maximizes a Student-t likelihood, admitting a hierarchical probabilistic interpretation with per-data-point latent variances. The authors provide a JAX implementation (Robusta-HMF) and demonstrate the method on simulated data and Gaia DR3 RVS spectra, identifying known anomalies (e.g., a Be-star binary) and subtle M-dwarf Ca II triplet emission. The abstract is transparent that identification accuracy depends on hyperparameter choice and claims these can be set by cross-validation. This review is based on the abstract only; the full text was not available for assessment.

Significance. The simultaneous treatment of heteroskedastic uncertainties, missing data, robustness, and anomaly flagging in a single framework is a practically useful contribution for astronomical spectroscopy, where all three issues arise routinely. The Student-t reweighting approach and its hierarchical model interpretation are standard statistical tools, but their assembly into a fast JAX implementation with practical guidance is valuable for the community. The Gaia DR3 application—particularly the recovery of M-dwarfs with subtle Ca II emission that would not be obvious by eye—demonstrates a concrete, falsifiable scientific use case. Credit is due for providing reproducible code and for the transparency regarding hyperparameter dependence.

major comments (2)
  1. The central practical claim is that RHMF simultaneously recovers a low-rank embedding AND flags anomalies. When outliers are structured (e.g., a coherent subgroup deviating along a direction that could be absorbed into a principal component), the Student-t reweighting faces an identifiability problem analogous to the rank-sparsity tradeoff in robust PCA (Candès et al. 2011). The abstract acknowledges hyperparameter sensitivity but does not indicate whether the full text addresses whether any hyperparameter setting can resolve structured-outlier ambiguity. The Gaia M-dwarf Ca II emission application involves outliers that may be structured in spectral space, making this concern directly relevant. The full text must explicitly discuss this identifiability limitation and, ideally, provide a test with structured outliers to show the boundary where the method fails.
  2. The claim that hyperparameters 'can be set reliably by cross-validation' is load-bearing for the anomaly-flagging capability. Without the full text, it is impossible to verify whether the cross-validation procedure is demonstrated across diverse data regimes or only on the specific simulations shown. The full text should specify the cross-validation protocol, show its performance across at least two qualitatively different outlier regimes (pointwise and structured), and discuss failure modes.
minor comments (2)
  1. Without the full text, it is not possible to verify the mathematical claims (Student-t likelihood equivalence, hierarchical model interpretation). These are standard results, but the full derivation should be checked during full review.
  2. The abstract mentions 'practical guidance for users' but does not specify its scope. The full text should include clear guidance on rank selection, convergence thresholds, and anomaly thresholding.

Circularity Check

0 steps flagged

No circularity detected: the method is defined against external benchmarks and standard statistical results, with no self-cited load-bearing chain.

full rationale

The abstract describes RHMF as an iterative reweighting algorithm that implicitly maximizes a Student-t likelihood with a hierarchical probabilistic interpretation. The Student-t connection to robust regression and latent-variance reweighting is a standard statistical result, not a self-cited circular dependency. The method is validated against external benchmarks: simulated data with known outliers, and Gaia DR3 RVS spectra where detected anomalies are checked against known objects (a Be star binary, M-dwarfs with Ca II triplet emission). No fitted parameter is renamed as a prediction, no uniqueness theorem is invoked, and no ansatz is smuggled through self-citation. The practical concern about cross-validation reliability and structured-outlier identifiability is a correctness risk, not a circularity issue. With only the abstract available, no load-bearing step reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 3 axioms · 0 invented entities

The paper introduces no new physical entities or mathematical objects beyond the method itself. The free parameters are standard for robust factor models. The axioms are domain assumptions about data noise structure and hyperparameter selection reliability. Without the full text, the exact parameter count and whether any are fitted to specific datasets cannot be determined.

free parameters (4)
  • Degrees of freedom of Student-t likelihood (inferred)
    Not stated in abstract, but Student-t based robust methods typically require a degrees-of-freedom parameter or a fixed value controlling robustness.
  • Number of latent factors (rank)
    Standard for any matrix factorization; likely chosen by cross-validation per the abstract's hyperparameter discussion.
  • Reweighting convergence threshold
    Iterative reweighting algorithms require stopping criteria; not mentioned in abstract.
  • Anomaly flagging threshold
    The abstract mentions automatic flagging of anomalies; the threshold for flagging is a hyperparameter the abstract says is set by cross-validation.
axioms (3)
  • domain assumption Data noise is well-approximated by a Student-t distribution or that the Student-t likelihood provides adequate robustness for the outlier regimes encountered.
    The method implicitly maximizes a Student-t likelihood; this is the core probabilistic assumption.
  • domain assumption Cross-validation on available data reliably selects hyperparameters for the anomaly detection task.
    The abstract explicitly claims hyperparameters 'can be set reliably by cross-validation,' which is a load-bearing assumption for practical use.
  • domain assumption The low-rank factorization model is appropriate for the data structure (i.e., the data genuinely has a low-dimensional embedding).
    Matrix factorization assumes the data matrix is well-approximated by a low-rank structure plus noise.

pith-pipeline@v1.1.0-glm · 4681 in / 2333 out tokens · 247288 ms · 2026-07-10T00:23:55.005180+00:00 · methodology

0 comments
read the original abstract

We present Robust Heteroskedastic Matrix Factorization (RHMF), a generalization of Principal Component Analysis (PCA) that is robust to outliers, handles per-feature uncertainties and missing data, and automatically flags per-feature and per-object anomalies. RHMF is useful both in recovering a low-dimensional embedding unspoiled by bad data or anomalies, and in identifying those anomalies. It utilises an iterative reweighting algorithm that implicitly maximizes a Student-t likelihood. This admits an equivalent probabilistic interpretation as fitting a hierarchical model with per-data-point latent variances. We deliver a fast JAX implementation, Robusta-HMF, and practical guidance for users. We demonstrate the ability of the model to identify and mitigate outliers of different classes. Identification accuracy is contingent on the choice of hyperparameters, but we show that these can be set reliably by cross-validation. We also apply RHMF to RVS spectra from Gaia DR3 to find main-sequence stars that are strange relative to their neighbors in color-magnitude space. We highlight specific examples, including a known binary hosting a Be star, and M-dwarfs with subtle emission in the Ca II triplet lines, indicative of accretion or magnetic activity, which would not be obvious to identify by eye.

Figures

Figures reproduced from arXiv: 2607.08081 by Andrew R. Casey, David W. Hogg, Hans-Walter Rix, Thomas Hilder.

Figure 1
Figure 1. Figure 1: Randomly selected noisy toy spectrum containing outlier absorptions lines, random outlier bad pixels, a pixel bad across many spectra, and a missing data segment. Top panel shows the data compared with the best-fit PCA (red, dashed-dotted), RPCA (blue, dotted) and RHMF (green, dashed) models. Middle panel shows the RHMF model-subtracted residuals, demonstrating that the model has ignored the outlier pixels… view at source ↗
Figure 2
Figure 2. Figure 2: Noisy toy spectra randomly selected from across the training and test sets, with the best-fit RHMF model predictions (black, dashed) overlaid. The top 3 are “normal” spectra (blue), while the bottom 3 are outlier spectra (orange). In the normal spectra, the model picks up on the common structure and ignores pixel-level outliers, while in the outlier spectra the model does not attempt to fit the high-freque… view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of the object-level weights w object i for the toy data, coloured by whether the spectrum is truly outlier (orange, hatched) or not (blue). The model clearly distinguishes the outlier spectra, which tend to have lower weights than normal spectra. The shown model has K = 5 and Q = 5. 0.5 1.0 2.0 3.0 4.0 5.0 10.0 Robust Scale Q 3 4 5 6 7 Rank K Cross-Validation −4 −2 0 log10 KL( p zkN (0, 1)) (L… view at source ↗
Figure 4
Figure 4. Figure 4: Top: Cross-validation scores (KL divergence from N (0, 1), Eq. (28)) for the grid of Q and K values. The score is minimized at K = 5 and Q = 5, which is the model shown in the other plots. Bottom left: F1 score for outlier identification on a per-object-per-pixel basis, based on a threshold of 0.5 on the robust weights w robust ij . Bottom right: F1 score for outlier identification on a per-object basis, b… view at source ↗
Figure 5
Figure 5. Figure 5: Gaia BP−RP color vs absolute G magnitude diagram, showing all stars in black. The subset we used for the analysis are colored by the bin they belong to. Each bin is labelled by an index 0 through 13, and the number of spectra in each bin is also shown. The bin edges overlap, and so many spectra belong to two bins, although this is not shown in the plot. in Gaia BP−RP color and nearly evenly spaced in G-ban… view at source ↗
Figure 6
Figure 6. Figure 6: Distribution of the object-level weights w object i for each bin, with a vertical grey dashed line showing the threshold of 0.5 for outlier identification. The number of identified outliers with w object i < 0.5, as well as the total number of spectra in each bin, is labelled next to the corresponding histogram. In the upper main sequence (closer to A and F-type stars; bins 0 through 5), the tail of the di… view at source ↗
Figure 7
Figure 7. Figure 7: Gaia/RVS spectrum (black) of Gaia DR3 1363284299777747584 (HD 162732; 88 Her) that we identified as an outlier (w object i = 9 × 10−3 ) in bin 0, shown with the RHMF reconstruction (green, dashed). This system is a suspected binary system that hosts a Be star with an equatorial disk of gas. in the data, and thus more spectra with low weights. We note, however, that this is not the only interpretation: the … view at source ↗
Figure 8
Figure 8. Figure 8: Gaia/RVS spectrum (black) of a more typical star (Gaia DR3 5309096898078973568) on the upper main sequence, shown with the good reconstruction (green, dashed). Unlike the outlier spectrum shown in [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Two M-dwarf outliers identified in bin 13: Gaia DR3 3136952686035250688 (top), and Gaia DR3 3195919254111314816 (bottom). Both show the model reconstructions in green (dashed), where the residual panels highlight emission in the Ca II triplet, consistent with stellar activity. The emission in the top panel is visibly obvious. The emission signature in lower panel is only discernible when compared to the mo… view at source ↗

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Works this paper leans on

76 extracted references · 76 canonical work pages · 13 internal anchors

  1. [1]

    and Morrell, Nidia I

    Abt, Helmut A. and Morrell, Nidia I. , year = 1995, month = jul, journal =. The. doi:10.1086/192182 , url =

    work page doi:10.1086/192182 1995

  2. [2]

    A Fast Iterative Robust Principal Component Analysis Method

    Aime Judicael Ouermi, Timbwaoga and Li, Jixian and Johnson, Chris R. , year = 2025, month = jun, journal =. A. doi:10.48550/arXiv.2506.16013 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2506.16013 2025

  3. [3]

    , year = 2012, journal =

    Amara, Adam and Quanz, Sascha P. , year = 2012, journal =

    work page 2012

  4. [4]

    Andrews, D. F. and Mallows, C. L. , year = 1974, month = sep, journal =. Scale. doi:10.1111/j.2517-6161.1974.tb00989.x , url =

    work page doi:10.1111/j.2517-6161.1974.tb00989.x 1974

  5. [5]

    The Astrophysical Journal , volume =

    The. The Astrophysical Journal , volume =

    work page

  6. [6]

    Principal Component Analysis with Noisy and/or Missing Data

    Bailey, Stephen , year = 2012, month = sep, journal =. Principal. doi:10.1086/668105 , url =. arXiv , keywords =:1208.4122 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  7. [7]

    Bayesian Theory , author =

    work page

  8. [8]

    Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset

    Bouwmans, Thierry and Sobral, Andrews and Javed, Sajid and Jung, Soon Ki and Zahzah, El-Hadi , year = 2015, month = nov, journal =. Decomposition into. doi:10.48550/arXiv.1511.01245 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1511.01245 2015

  9. [9]

    Accelerated Alternating Projections for Robust Principal Component Analysis

    Cai, HanQin and Cai, Jian-Feng and Wei, Ke , year = 2017, month = nov, journal =. Accelerated. doi:10.48550/arXiv.1711.05519 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1711.05519 2017

  10. [10]

    Cai, HanQin and Hamm, Keaton and Huang, Longxiu and Li, Jiaqi and Wang, Tao , year = 2021, journal =. Rapid. doi:10.1109/LSP.2020.3044130 , url =

    work page doi:10.1109/lsp.2020.3044130 2021

  11. [11]

    Cai, HanQin and Liu, Jialin and Yin, Wotao , year = 2021, month = oct, number =. Learned. doi:10.48550/arXiv.2110.05649 , url =. arXiv , keywords =:2110.05649 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2110.05649 2021

  12. [12]

    Robust Principal Component Analysis? , author =. J. ACM , volume =. doi:10.1145/1970392.1970395 , url =

    work page doi:10.1145/1970392.1970395

  13. [13]

    and Hogg, David W

    Casey, Andrew R. and Hogg, David W. and Ness, Melissa and Rix, Hans-Walter and Ho, Anna Q Y and Gilmore, Gerry , year = 2016, journal =. The

    work page 2016

  14. [14]

    and Wheeler, Adam and Bedell, Megan and Hogg, David W

    Casey, Andrew R. and Wheeler, Adam and Bedell, Megan and Hogg, David W. and Saydjari, Andrew and Zhao, Lily , year = 2026, month = feb, journal =. A. doi:10.3847/1538-4357/ae3bd6 , url =

    work page doi:10.3847/1538-4357/ae3bd6 2026

  15. [15]

    Robust Principal Components by Casewise and Cellwise Weighting

    Centofanti, Fabio and Hubert, Mia and Rousseeuw, Peter J. , year = 2024, month = aug, journal =. Robust. doi:10.48550/arXiv.2408.13596 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2408.13596 2024

  16. [16]

    Gaia Collaboration and Prusti, T. and. The. Astronomy and Astrophysics , volume =. doi:10.1051/0004-6361/201629272 , url =

    work page doi:10.1051/0004-6361/201629272

  17. [17]

    Python and

    Collette, Andrew , year = 2013, publisher =. Python and

    work page 2013

  18. [18]

    Weighted principal component analysis: a weighted covariance eigendecomposition approach

    Delchambre, Ludovic , year = 2015, month = feb, journal =. Weighted Principal Component Analysis: A Weighted Covariance Eigendecomposition Approach , shorttitle =. doi:10.1093/mnras/stu2219 , url =. arXiv , keywords =:1412.4533 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  19. [19]

    Conjugate

    Diaconis, Persi and Ylvisaker, Donald , year = 1979, journal =. Conjugate. 2958808 , eprinttype =

    work page 1979

  20. [20]

    Psychometrika , volume =

    The Approximation of One Matrix by Another of Lower Rank , author =. Psychometrika , volume =. doi:10.1007/BF02288367 , url =

    work page doi:10.1007/bf02288367

  21. [21]

    The Astrophysical Journal , volume =

    S4: A Spatial-Spectral Model for Speckle Suppression , author =. The Astrophysical Journal , volume =

    work page

  22. [22]

    Dganguli/Robust-Pca , author =

    work page

  23. [23]

    Gomez Gonzalez, C. A. and Absil, O. and Absil, P.-A. and Van Droogenbroeck, M. and Mawet, D. and Surdej, J. , year = 2016, month = may, journal =. Low-Rank plus Sparse Decomposition for Exoplanet Detection in Direct-Imaging. doi:10.1051/0004-6361/201527387 , url =

    work page doi:10.1051/0004-6361/201527387 2016

  24. [24]

    , year = 2019, month = jan, journal =

    Guan, Naiyang and Liu, Tongliang and Zhang, Yangmuzi and Tao, Dacheng and Davis, Larry S. , year = 2019, month = jan, journal =. Truncated. doi:10.1109/TPAMI.2017.2777841 , url =. arXiv , keywords =:1906.00495 , primaryclass =

    work page doi:10.1109/tpami.2017.2777841 2019

  25. [25]

    Foreground Detection via Robust Low Rank Matrix Factorization Including Spatial Constraint with

    Guyon, Charles and Bouwmans, Thierry and Zahzah, El-Hadi , year = 2012, month = nov, pages =. Foreground Detection via Robust Low Rank Matrix Factorization Including Spatial Constraint with. Proceedings of the 21st

    work page 2012

  26. [26]

    and Martinsson, P

    Halko, N. and Martinsson, P. G. and Tropp, J. A. , year = 2011, journal =. Finding Structure with Randomness:. doi:10.1137/090771806 , url =. https://doi.org/10.1137/090771806 , pages =

    work page doi:10.1137/090771806 2011

  27. [27]

    and Millman, K

    Harris, Charles R. and Millman, K. Jarrod and. Array Programming with. Nature , volume =

    work page

  28. [28]

    He, Lingyu and Yang, Yanrong and Zhang, Bo , year = 2022, month = apr, number =. Robust. doi:10.48550/arXiv.2204.01042 , url =. arXiv , keywords =:2204.01042 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2204.01042 2022

  29. [29]

    Hilder, Tom , year = 2026, month = feb, url =

    work page 2026

  30. [30]

    Understanding Robust and Exploratory Data Analysis , author =

    work page

  31. [31]

    and Bovy, Jo and Lang, Dustin , year = 2010, journal =

    Hogg, David W. and Bovy, Jo and Lang, Dustin , year = 2010, journal =. Data Analysis Recipes:

    work page 2010

  32. [32]

    Communications in Statistics -- Theory and Methods , volume =

    Robust Regression Using Iteratively Reweighted Least-Squares , author =. Communications in Statistics -- Theory and Methods , volume =

    work page

  33. [33]

    Journal of Educational Psychology , volume =

    Analysis of a Complex of Statistical Variables into Principal Components , author =. Journal of Educational Psychology , volume =. doi:10.1037/h0071325 , abstract =

    work page doi:10.1037/h0071325

  34. [34]

    Theory of Stellar Atmospheres:

    Hubeny, Ivan and Mihalas, Dimitri , year = 2014, series =. Theory of Stellar Atmospheres:

    work page 2014

  35. [35]

    , year = 2007, journal =

    Hunter, John D. , year = 2007, journal =. Matplotlib:

    work page 2007

  36. [36]

    Bradbury, James and Frostig, Roy and Hawkins, Peter and Johnson, Matthew James and Leary, Chris and Maclaurin, Dougal and Necula, George and Paszke, Adam and VanderPlas, Jake and

    work page

  37. [37]

    and Sartoretti, P

    Katz, D. and Sartoretti, P. and Guerrier, A. and Panuzzo, P. and Seabroke, G. M. and Th. Gaia. Astronomy & Astrophysics , volume =. doi:10.1051/0004-6361/202244220 , url =

    work page doi:10.1051/0004-6361/202244220

  38. [38]

    Ke, Qifa and Kanade, Takeo , year = 2005, month = jun, series =. Robust. Proceedings of the 2005. doi:10.1109/CVPR.2005.309 , url =

    work page doi:10.1109/cvpr.2005.309 2005

  39. [39]

    Equinox:

    Kidger, Patrick and Garcia, Cristian , year = 2021, eprint =. Equinox:

    work page 2021

  40. [40]

    and Gies, Douglas R

    Klement, Robert and Schaefer, Gail H. and Gies, Douglas R. and Wang, Luqian and Baade, Dietrich and Rivinius, Thomas and Gallenne, Alexandre and Carciofi, Alex C. and Monnier, John D. and M. Interferometric. The Astrophysical Journal , volume =. doi:10.3847/1538-4357/ac4266 , url =

    work page doi:10.3847/1538-4357/ac4266

  41. [41]

    and Leibler, R

    Kullback, S. and Leibler, R. A. , year = 1951, month = mar, journal =. On. doi:10.1214/aoms/1177729694 , url =

    work page doi:10.1214/aoms/1177729694 1951

  42. [42]

    Lakshminarayanan, Balaji and Bouchard, Guillaume and Archambeau, Cedric , year = 2011, month = jun, pages =. Robust. Proceedings of the

    work page 2011

  43. [43]

    and Little, Roderick J

    Lange, Kenneth L. and Little, Roderick J. A. and Taylor, Jeremy M. G. , year = 1989, month = dec, journal =. Robust. doi:10.1080/01621459.1989.10478852 , url =

    work page doi:10.1080/01621459.1989.10478852 1989

  44. [44]

    Nature , volume =

    Learning the Parts of Objects by Non-Negative Matrix Factorization , author =. Nature , volume =

    work page

  45. [45]

    Nouvelles M\'ethodes Pour La D\'etermination Des Orbites Des Com\`etes , author =

    work page

  46. [46]

    Lin, Zhouchen and Chen, Minming and Ma, Yi , year = 2010, month = sep, journal =. The. doi:10.48550/arXiv.1009.5055 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1009.5055 2010

  47. [47]

    Magnier,. Pan-. The Astrophysical Journal Supplement Series , volume =. doi:10.3847/1538-4365/abb82a , url =

    work page doi:10.3847/1538-4365/abb82a

  48. [48]

    UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

    McInnes, Leland and Healy, John and Melville, James , year = 2018, journal =. doi:10.48550/arXiv.1802.03426 , archiveprefix =. 1802.03426 , pages =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1802.03426 2018

  49. [49]

    and Rix, Hans-Walter and Ho, Anna Y

    Ness, Melissa and Hogg, David W. and Rix, Hans-Walter and Ho, Anna Y. Q. and Zasowski, Gail , year = 2015, journal =. The

    work page 2015

  50. [50]

    Advances in Neural Information Processing Systems , author =

    Non-Convex Robust. Advances in Neural Information Processing Systems , author =

    work page

  51. [51]

    doi:10.1080/14786440109462720 , langid =

    Pearson, Karl , year = 1901, month = nov, journal =. doi:10.1080/14786440109462720 , url =

    work page doi:10.1080/14786440109462720 1901

  52. [52]

    and Khlebnikov, Mikhail V

    Polyak, Boris T. and Khlebnikov, Mikhail V. , year = 2017, month = jul, journal =. Robust. doi:10.1016/j.ifacol.2017.08.585 , url =

    work page doi:10.1016/j.ifacol.2017.08.585 2017

  53. [53]

    Monthly Notices of the Royal Astronomical Society , volume =

    The Dimensionality of Stellar Chemical Space Using Spectra from the. Monthly Notices of the Royal Astronomical Society , volume =. doi:10.1093/mnras/stx3198 , url =

    work page doi:10.1093/mnras/stx3198

  54. [54]

    Re Fiorentin, P. and. Estimation of Stellar Atmospheric Parameters from. Astronomy and Astrophysics , volume =

    work page

  55. [55]

    Astronomy & Astrophysics , volume =

    Bright Be-Shell Stars , author =. Astronomy & Astrophysics , volume =. doi:10.1051/0004-6361:20053008 , adsnote =

    work page doi:10.1051/0004-6361:20053008

  56. [56]

    Fast Principal Component Pursuit via Alternating Minimization , booktitle =

    Rodr. Fast Principal Component Pursuit via Alternating Minimization , booktitle =. doi:10.1109/ICIP.2013.6738015 , url =

    work page doi:10.1109/icip.2013.6738015 2013

  57. [57]

    and Giampouras, Paris V

    Rontogiannis, Athanasios A. and Giampouras, Paris V. and Koutroumbas, Konstantinos D. , year = 2020, journal =. Online. doi:10.1109/LSP.2020.3011896 , url =

    work page doi:10.1109/lsp.2020.3011896 2020

  58. [58]

    Neural Computation , volume =

    Nonlinear Component Analysis as a Kernel Eigenvalue Problem , author =. Neural Computation , volume =

    work page

  59. [59]

    and Varoquaux, G

    Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V. and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P. and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and Cournapeau, D. and Brucher, M. and Perrot, M. and Duchesnay, E. , year = 2011, journal =. Scikit-Learn:

    work page 2011

  60. [60]

    Detection and Characterization of Exoplanets and Disks Using Projections on

    Soummer, R. Detection and Characterization of Exoplanets and Disks Using Projections on. The Astrophysical Journal Letters , volume =

    work page

  61. [61]

    Robust Principal Component Analysis Based on Low-Rank and Block-Sparse Matrix Decomposition , booktitle =

    Tang, Gongguo and Nehorai, Arye , year = 2011, month = mar, pages =. Robust Principal Component Analysis Based on Low-Rank and Block-Sparse Matrix Decomposition , booktitle =. doi:10.1109/CISS.2011.5766144 , url =

    work page doi:10.1109/ciss.2011.5766144 2011

  62. [62]

    Ting, Yuan-Sen and Conroy, Charlie and Rix, Hans-Walter and Cargile, Phillip , year = 2019, journal =. The. doi:10.3847/1538-4357/ab2331 , archiveprefix =. 1804.01530 , pages =

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  63. [63]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume =

    Probabilistic Principal Component Analysis , author =. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume =

    work page

  64. [64]

    and Hogg, David W

    Tsalmantza, P. and Hogg, David W. , year = 2012, month = jul, journal =. A. doi:10.1088/0004-637X/753/2/122 , url =

    work page doi:10.1088/0004-637x/753/2/122 2012

  65. [65]

    and Brown, A

    Vallenari, A. and Brown, A. G. A. and Prusti, T. and de Bruijne, J. H. J. and Arenou, F. and Babusiaux, C. and Biermann, M. and Creevey, O. L. and Ducourant, C. and Evans, D. W. and Eyer, L. and Guerra, R. and Hutton, A. and Jordi, C. and Klioner, S. A. and Lammers, U. L. and Lindegren, L. and Luri, X. and Mignard, F. and Panem, C. and Pourbaix, D. and Ra...

    work page doi:10.1051/0004-6361/202243940

  66. [66]

    Vaswani, Namrata and Bouwmans, Thierry and Javed, Sajid and Narayanamurthy, Praneeth , year = 2018, month = jul, journal =. Robust. doi:10.1109/MSP.2018.2826566 , url =

    work page doi:10.1109/msp.2018.2826566 2018

  67. [67]

    and Haberland, Matt and Reddy, Tyler and Cournapeau, David and Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and Bright, Jonathan and

    Virtanen, Pauli and Gommers, Ralf and Oliphant, Travis E. and Haberland, Matt and Reddy, Tyler and Cournapeau, David and Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and Bright, Jonathan and. Nature Methods , volume =

    work page

  68. [68]

    West, Mike , year = 1984, month = jul, journal =. Outlier. doi:10.1111/j.2517-6161.1984.tb01317.x , url =

    work page doi:10.1111/j.2517-6161.1984.tb01317.x 1984

  69. [69]

    Biometrika , volume =

    On Scale Mixtures of Normal Distributions , author =. Biometrika , volume =. doi:10.1093/biomet/74.3.646 , url =

    work page doi:10.1093/biomet/74.3.646

  70. [70]

    A Unified

    Wipf, David and Nagarajan, Srikantan , year = 2009, month = feb, journal =. A Unified. doi:10.1016/j.neuroimage.2008.02.059 , url =

    work page doi:10.1016/j.neuroimage.2008.02.059 2009

  71. [71]

    Local Principal Component Pursuit for Nonlinear Datasets , booktitle =

    Wohlberg, Brendt and Chartrand, Rick and Theiler, James , year = 2012, month = mar, pages =. Local Principal Component Pursuit for Nonlinear Datasets , booktitle =. doi:10.1109/ICASSP.2012.6288776 , url =

    work page doi:10.1109/icassp.2012.6288776 2012

  72. [72]

    Wright, John and Ganesh, Arvind and Rao, Shankar and Peng, Yigang and Ma, Yi , year = 2009, volume =. Robust. Advances in

    work page 2009

  73. [73]

    Yin, Lei and Parekh, Ankit and Selesnick, Ivan , year = 2019, month = may, journal =. Stable. doi:10.1109/TSP.2019.2907264 , url =

    work page doi:10.1109/tsp.2019.2907264 2019

  74. [74]

    Heteroskedastic PCA: Algorithm, Optimality, and Applications

    Zhang, Anru R. and Cai, T. Tony and Wu, Yihong , year = 2018, month = oct, journal =. Heteroskedastic. doi:10.48550/arXiv.1810.08316 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1810.08316 2018

  75. [75]

    and Hogg, David W

    Zhao, Lily L. and Hogg, David W. and Bedell, Megan and Fischer, Debra A. , year = 2021, month = jan, journal =. Excalibur:. doi:10.3847/1538-3881/abd105 , url =

    work page doi:10.3847/1538-3881/abd105 2021

  76. [76]

    Zhou, Zihan and Li, Xiaodong and Wright, John and Candes, Emmanuel and Ma, Yi , year = 2010, month = jan, journal =. Stable. doi:10.48550/arXiv.1001.2363 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1001.2363 2010