pith. sign in

arxiv: 2604.08838 · v1 · submitted 2026-04-10 · 📡 eess.SP

Exploring Bounded Component Analysis Using an ell_infty Norm Criterion

Renan D. B. Brotto , Kenji Nose-Filho , Jo\~ao M. T. Romano This is my paper

Pith reviewed 2026-05-10 18:13 UTC · model grok-4.3

classification 📡 eess.SP
keywords blind source separationbounded component analysisl infinity normantisparse sourcesGivens rotationscontrast functionindependent component analysis
0
0 comments X

The pith

Minimizing the sum of the ℓ∞ norms of recovered sources estimates the separation matrix for antisparse bounded signals.

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

The paper shows that the mixing of bounded sources by any non-identity matrix with unit Frobenius norm increases their ℓ∞ norms, so the separation matrix can be recovered by minimizing the sum of those norms. This contrast function is implemented by first applying principal component analysis and then optimizing over Givens rotations for independent sources, or by using Givens rotations alone when the sources are correlated and the mixing is a rotation. Numerical tests on three families of bounded signals demonstrate that the approach outperforms a competing algorithm.

Core claim

The authors prove that for bounded sources the ℓ∞ norm grows under mixing by any unitary-Frobenius-norm matrix other than the identity, and therefore the separation matrix is the minimizer of the sum of the ℓ∞ norms of the estimated sources. This criterion is realized by a PCA stage followed by Givens-rotation optimization for independent sources and by direct Givens optimization when the mixing matrix itself is a rotation.

What carries the argument

The sum of the ℓ∞ norms of the estimated sources, minimized through PCA preprocessing and Givens-rotation search.

If this is right

  • Independent bounded sources are separated by PCA followed by Givens-rotation optimization.
  • Correlated bounded sources mixed by a rotation matrix are separated by Givens-rotation optimization alone.
  • The ℓ∞ norm acts as an effective contrast function for antisparse bounded sources and outperforms a state-of-the-art method in simulations.
  • The theoretical norm-increase property supplies a justification for using this criterion in blind source separation.

Where Pith is reading between the lines

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

  • The same norm-increase argument could be tested for other peak-based norms or for sources with different amplitude bounds.
  • The method’s low computational cost after PCA makes it attractive for real-time separation of bounded sensor or communication signals.
  • Hybrid criteria that combine the ℓ∞ term with sparsity or independence penalties might handle mixtures containing both bounded and sparse components.
  • Direct application to real clipped audio or saturated sensor recordings would provide an immediate test of practical utility.

Load-bearing premise

Any mixing matrix with unit Frobenius norm other than the identity strictly increases the ℓ∞ norm of bounded sources.

What would settle it

A concrete counterexample of a non-identity unitary-Frobenius-norm matrix that leaves the ℓ∞ norm of some bounded sources unchanged, or a simulation in which the proposed minimization fails to recover the sources.

Figures

Figures reproduced from arXiv: 2604.08838 by Jo\~ao M. T. Romano, Kenji Nose-Filho, Renan D. B. Brotto.

Figure 1
Figure 1. Figure 1: Illustration of the Extreme Points Condition of Cruces’ work [19] for the case of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sources and Mixtures Joint Support. Originally, the source samples lay inside the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sufficiently Spread Sources Condition. The sources support, represented by the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: SER Comparative - Uniform Distribution. Considering 2, 5 and 10 sources, we [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: SER Comparative - Multimodal Distribution. Considering 2, 5 and 10 sources, we [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results of a single simulation in a noiseless case. From left to right we present the [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: PSNR x SNR. We took the mean PNSR from 10 Monte Carlo simulations, consid [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Correlation and Noise Level Analysis. Considering 2, 5 and 10 sources, we compared [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
read the original abstract

In this paper we propose a new criterion for the Blind Source Separation (BSS) of antisparse bounded sources, based on the sum of the $\ell_\infty$-norm of the sources. Based on the observation that the mixing process of bounded sources with any mixing matrix with unitary Frobenius norm will increase the $\ell_\infty$-norm of the sources, unless it is the identity matrix, the minimization of the sum of the $\ell_\infty$-norm of the sources can be used for the estimation of a separation matrix. To that, a Principle Component Analysis technique followed by a Givens Rotations based optimization method can be used for the separation of independent bounded sources. Also, the Givens Rotations based optimization method can be used for the separation of correlated bounded sources mixed by a rotation matrix. We theoretically analyze the proposed criterion and assess its performance through numerical simulations involving three distinct types of bounded signals. Our theoretical and experimental findings underscore the efficacy of the $\ell_\infty$ norm as a suitable contrast function for antisparse bounded sources, showcasing its superior performance relative to a state-of-the-art algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a new contrast function for blind source separation (BSS) of antisparse bounded sources based on minimizing the sum of the ℓ∞-norms of the recovered sources. The approach rests on the observation that mixing such sources by any matrix A with unit Frobenius norm strictly increases this sum unless A is the identity; the authors therefore advocate PCA followed by Givens-rotation optimization for independent sources and Givens rotations alone for correlated sources mixed by a rotation matrix. Theoretical analysis of the criterion is provided together with numerical simulations on three families of bounded signals, with the claim that the ℓ∞ criterion outperforms a state-of-the-art algorithm.

Significance. If the central observation can be placed on a firm footing by adding the necessary restrictions on A (orthogonality or full rank), the work would supply a simple, parameter-light contrast function for bounded-component analysis that is directly motivated by the geometry of the ℓ∞ ball. The combination of an explicit theoretical claim and comparative simulations is a positive feature; the method’s reliance on Givens rotations also offers a computationally attractive implementation path.

major comments (2)
  1. [Abstract] Abstract (and the corresponding theoretical section): the claim that “the mixing process of bounded sources with any mixing matrix with unitary Frobenius norm will increase the ℓ∞-norm of the sources, unless it is the identity matrix” is false in general. Counter-example: let n=2 and A=[[1,0],[0,0]] (‖A‖_F=1). For antisparse sources attaining all sign combinations with ‖s_i‖_∞=1, the mixed signals satisfy ‖x_1‖_∞=1 and ‖x_2‖_∞=0, so the sum equals 1, which is strictly smaller than √2 obtained by the scaled identity of the same Frobenius norm. The inequality therefore requires unstated restrictions (e.g., A orthogonal or full-rank); without them the proposed sum-of-ℓ∞ criterion is not guaranteed to be a valid contrast function whose minimum occurs only at the separating matrix. This is load-bearing for the central claim.
  2. [Theoretical analysis] Theoretical analysis section: the derivation that the minimum of the sum-of-ℓ∞ criterion occurs at the identity (or separating matrix) must explicitly list the assumptions on the mixing matrix under which the observation holds. The current statement “any mixing matrix with unitary Frobenius norm” is too broad and must be corrected or qualified before the contrast-function property can be asserted.
minor comments (2)
  1. [Numerical simulations] Simulations: the abstract states that three distinct types of bounded signals were used and that the method outperforms a state-of-the-art algorithm, yet no details are given on the number of Monte-Carlo trials, data-generation procedure, error-bar reporting, or exclusion rules. These omissions make the reported superiority difficult to assess.
  2. [Method description] Notation and scope: clarify whether the sources are assumed statistically independent throughout or whether the Givens-rotation procedure is intended only for the rotation-matrix case; the transition between the two regimes should be stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the insightful comments and the counterexample provided. We agree that the original statement of the central observation is too general and requires explicit restrictions on the mixing matrix to hold. Our method relies on orthogonal transformations (via PCA and Givens rotations), so we will revise the manuscript to qualify the claim accordingly. This addresses the major concerns while preserving the core contribution.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the corresponding theoretical section): the claim that “the mixing process of bounded sources with any mixing matrix with unitary Frobenius norm will increase the ℓ∞-norm of the sources, unless it is the identity matrix” is false in general. Counter-example: let n=2 and A=[[1,0],[0,0]] (‖A‖_F=1). For antisparse sources attaining all sign combinations with ‖s_i‖_∞=1, the mixed signals satisfy ‖x_1‖_∞=1 and ‖x_2‖_∞=0, so the sum equals 1, which is strictly smaller than √2 obtained by the scaled identity of the same Frobenius norm. The inequality therefore requires unstated restrictions (e.g., A orthogonal or full-rank); without them the proposed sum-of-ℓ∞ criterion is not guaranteed to be a valid contrast function whose minimum occurs only at the separating matrix. This is load-bearing for the central claim.

    Authors: We acknowledge the validity of this counterexample and agree that the claim as originally stated does not hold for arbitrary matrices with unit Frobenius norm. In the proposed approach, PCA is first applied to the observations, which effectively orthogonalizes the mixing process (assuming the sources are bounded and the data is whitened). For the independent sources case, the subsequent optimization uses Givens rotations, which are orthogonal. For correlated sources, the mixing is assumed to be a rotation matrix, hence orthogonal. Under the assumption that A is an orthogonal matrix with ||A||_F = 1 (after suitable normalization of the sources or data), the sum of ℓ∞ norms is indeed minimized when A is the identity (up to sign flips and permutations, which are handled in BSS). We will revise the abstract to reflect this restriction and clarify the context of the method. revision: yes

  2. Referee: [Theoretical analysis] Theoretical analysis section: the derivation that the minimum of the sum-of-ℓ∞ criterion occurs at the identity (or separating matrix) must explicitly list the assumptions on the mixing matrix under which the observation holds. The current statement “any mixing matrix with unitary Frobenius norm” is too broad and must be corrected or qualified before the contrast-function property can be asserted.

    Authors: We agree that the theoretical analysis section must explicitly state the assumptions. We will update the section to specify that the observation holds for orthogonal mixing matrices A with unit Frobenius norm (i.e., A^T A = I after normalization). The derivation will be adjusted to prove the minimum under this condition, which aligns with the algorithmic implementation using PCA and Givens rotations. This qualification ensures the contrast function property is correctly asserted. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central derivation starts from an explicit observation about the effect of mixing on the sum of ℓ∞-norms for bounded sources under unitary-Frobenius-norm matrices, then proposes minimization of that sum as a contrast function for separation. This observation is presented as an independent theoretical property that the authors analyze and validate separately; the subsequent PCA-plus-Givens procedure is offered only as an algorithmic realization, not as a redefinition or statistical fit of the same quantity. No equations reduce the claimed contrast property to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. The derivation chain therefore remains self-contained against external benchmarks and does not collapse by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on one domain assumption about norm increase under mixing; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Mixing bounded sources with any matrix of unitary Frobenius norm increases the ℓ∞-norm of the sources unless the matrix is the identity.
    This observation, stated in the abstract, underpins the entire minimization criterion.

pith-pipeline@v0.9.0 · 5510 in / 1295 out tokens · 80927 ms · 2026-05-10T18:13:52.511060+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    Comon, C

    P. Comon, C. Jutten (Eds.), Handbook of blind source seapartion, Academic Press, 2010. 19

    work page 2010

  2. [2]

    D. G. Fantinato, L. T. Duarte, Y. Deville, R. Attux, C. Jutten, A. Neves, A second- order statistics method for blind source separation in post-nonlinear mixtures, Signal Processing 155 (2019) 63–72

    work page 2019

  3. [3]

    Hyvarinen, J

    A. Hyvarinen, J. Karhunen, E. Oja, Independent component analysis, John Wiley & Sons, 2001

    work page 2001

  4. [4]

    C. P. Moraes, D. G. Fantinato, A. Neves, Epanechnikov kernel for pdf estimation applied to equalization and blind source separation, Signal Processing 189 (2021) 108251

    work page 2021

  5. [5]

    Friman, M

    O. Friman, M. Borga, P. Lundberg, H. Knutsson, Exploratory fmri anal- ysis by autocorrelation maximization, NeuroImage 16 (2) (2002) 454–464. doi:https://doi.org/10.1006/nimg.2002.1067. URLhttps://www.sciencedirect.com/science/article/pii/S1053811902910670

    work page doi:10.1006/nimg.2002.1067 2002

  6. [6]

    Bobin, J

    J. Bobin, J. Rapin, A. Larue, J.-L. Starck, Sparsity and adaptivity for the blind sep- aration of partially correlated sources, IEEE Transactions on Signal Processing 63 (5) (2015) 1199–1213. doi:10.1109/TSP.2015.2391071

    work page doi:10.1109/tsp.2015.2391071 2015

  7. [7]

    Wu, L.-L

    X. Wu, L.-L. Zeng, H. Shen, M. Li, Y. an Hu, D. Hu, Blind source separation of functional mri scans of the human brain based on canonical correlation analysis, Neurocomputing 269 (2017) 220–225. doi:https://doi.org/10.1016/j.neucom.2017.01.106. URLhttps://www.sciencedirect.com/science/article/pii/S0925231217309967

    work page doi:10.1016/j.neucom.2017.01.106 2017

  8. [8]

    Y. Xie, K. Xie, S. Xie, Underdetermined blind separation of source us- ing lp-norm diversity measures, Neurocomputing 411 (2020) 259–267. doi:https://doi.org/10.1016/j.neucom.2020.06.029. URLhttps://www.sciencedirect.com/science/article/pii/S0925231220310006

    work page doi:10.1016/j.neucom.2020.06.029 2020

  9. [9]

    Gribonval, S

    R. Gribonval, S. Lesage, A survey of sparse component analysis for blind source separa- tion: principles, perspectives, and new challenges, in: Proceedings of the 2006 ESANN, Bruges, Belgium, 2006, pp. 323–330. 20

    work page 2006

  10. [10]

    Georgiev, F

    P. Georgiev, F. Theis, A. Cichocki, Sparse component analysis and blind source separa- tion of underdetermined mixtures, IEEE Transactions on Neural Networks 16 (4) (2005) 992–996. doi:10.1109/TNN.2005.849840

    work page doi:10.1109/tnn.2005.849840 2005

  11. [11]

    Movahedi Naini, G

    F. Movahedi Naini, G. Hosein Mohimani, M. Babaie-Zadeh, C. Jutten, Esti- mating the mixing matrix in sparse component analysis (sca) based on par- tial k-dimensional subspace clustering, Neurocomputing 71 (10) (2008) 2330– 2343, neurocomputing for Vision Research Advances in Blind Signal Processing. doi:https://doi.org/10.1016/j.neucom.2007.07.035

    work page doi:10.1016/j.neucom.2007.07.035 2008

  12. [12]

    Elad, Sparse and Redundant Representations - From Theory to Applications in Signal and Image Processing, 1st Edition, Springer, New York, NY, 2010

    M. Elad, Sparse and Redundant Representations - From Theory to Applications in Signal and Image Processing, 1st Edition, Springer, New York, NY, 2010

    work page 2010

  13. [13]

    L. T. Duarte, R. Suyama, R. Attux, J. M. T. Romano, C. Jutten, Blind extraction of sparse components based onl 0-norm minimization, in: Proceedings of the 2011 SSP, 2011, pp. 617–620. doi:10.1109/SSP.2011.5967775

    work page doi:10.1109/ssp.2011.5967775 2011

  14. [14]

    Anemuller, Maximization of component disjointness: a criterion for blind source sep- aration, in: Independent Component Analysis, Springer Berlin Heidelberg, 2007

    J. Anemuller, Maximization of component disjointness: a criterion for blind source sep- aration, in: Independent Component Analysis, Springer Berlin Heidelberg, 2007

    work page 2007

  15. [15]

    Nose-Filho, L

    K. Nose-Filho, L. T. Duarte, J. M. T. Romano, On disjoint component analysis, in: P. Tichavsk´ y, M. Babaie-Zadeh, O. J. Michel, N. Thirion-Moreau (Eds.), Latent Variable Analysis and Signal Separation, Springer International Publishing, Cham, 2017, pp. 519– 528

    work page 2017

  16. [16]

    Cichocki, R

    A. Cichocki, R. Zdunek, S. Amari, New algorithms for non-negative matrix factoriza- tion in applications to blind source separation, in: 2006 IEEE International Confer- ence on Acoustics Speech and Signal Processing Proceedings, Vol. 5, 2006, pp. V–V. doi:10.1109/ICASSP.2006.1661352. 21

    work page doi:10.1109/icassp.2006.1661352 2006

  17. [17]

    Cichocki, R

    A. Cichocki, R. Zdunek, A. H. Phan, S. ichi Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation, 1st Edition, Wiley, 2009

    work page 2009

  18. [18]

    Hsieh, J.-T

    H.-L. Hsieh, J.-T. Chien, A new nonnegative matrix factorization for independent com- ponent analysis, in: 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, 2010, pp. 2026–2029. doi:10.1109/ICASSP.2010.5494945

    work page doi:10.1109/icassp.2010.5494945 2010

  19. [19]

    Cruces, Bounded component analysis of linear mixtures: A criterion of minimum convex perimeter, IEEE Transactions on Signal Processing 58 (4) (2010) 2141–2154

    S. Cruces, Bounded component analysis of linear mixtures: A criterion of minimum convex perimeter, IEEE Transactions on Signal Processing 58 (4) (2010) 2141–2154

    work page 2010

  20. [20]

    A. T. Erdogan, A class of bounded component analysis algorithms for the separation of both independent and dependent sources, IEEE Transactions on Signal Processing 61 (22) (2013) 5730–5743. doi:10.1109/TSP.2013.2280115

    work page doi:10.1109/tsp.2013.2280115 2013

  21. [21]

    H. A. Inan, A. T. Erdogan, An extended family of bounded component analysis algo- rithms, in: 2014 48th Asilomar Conference on Signals, Systems and Computers, 2014, pp. 442–445. doi:10.1109/ACSSC.2014.7094481

    work page doi:10.1109/acssc.2014.7094481 2014

  22. [22]

    W. S. B. Ouedraogo, A. Souloumiac, M. Ja¨ ıdane, C. Jutten, Non-negative blind source separation algorithm based on minimum aperture simplicial cone, IEEE Transactions on Signal Processing 62 (2) (2014) 376–389. doi:10.1109/TSP.2013.2287683

    work page doi:10.1109/tsp.2013.2287683 2014

  23. [23]

    Boulais, Y

    A. Boulais, Y. Deville, O. Bern, A geometrical blind separation method for unconstrained-sum locally dominant sources, in: 2015 IEEE International Workshop of Electronics, Control, Measurement, Signals and their Application to Mechatronics (ECMSM), 2015, pp. 1–6. doi:10.1109/ECMSM.2015.7208711

    work page doi:10.1109/ecmsm.2015.7208711 2015

  24. [24]

    Cruces, I

    S. Cruces, I. Dur´ an-D´ ıaz, The minimum risk principle that underlies the criteria of bounded component analysis, IEEE Transactions on Neural Networks and Learning Systems 26 (5) (2015) 964–981. doi:10.1109/TNNLS.2014.2329318. 22

    work page doi:10.1109/tnnls.2014.2329318 2015

  25. [25]

    Cruces, Bounded component analysis of noisy underdetermined and overdeter- mined mixtures, IEEE Transactions on Signal Processing 63 (9) (2015) 2279–2294

    S. Cruces, Bounded component analysis of noisy underdetermined and overdeter- mined mixtures, IEEE Transactions on Signal Processing 63 (9) (2015) 2279–2294. doi:10.1109/TSP.2015.2404312

    work page doi:10.1109/tsp.2015.2404312 2015

  26. [26]

    H. A. Inan, A. T. Erdogan, Convolutive bounded component analysis algorithms for independent and dependent source separation, IEEE Transactions on Neural Networks and Learning Systems 26 (4) (2015) 697–708. doi:10.1109/TNNLS.2014.2320817

    work page doi:10.1109/tnnls.2014.2320817 2015

  27. [27]

    H. A. Inan, A. T. Erdogan, A convolutive bounded component analysis framework for potentially nonstationary independent and/or dependent sources, IEEE Transactions on Signal Processing 63 (1) (2015) 18–30. doi:10.1109/TSP.2014.2367472

    work page doi:10.1109/tsp.2014.2367472 2015

  28. [28]

    Mansour, N

    A. Mansour, N. Ohnishi, C. Puntonet, Blind multiuser separation of instantaneous mix- ture algorithm based on geometrical concepts, Signal Processing 82 (8) (2002) 1155–1175. doi:https://doi.org/10.1016/S0165-1684(02)00250-5. URLhttps://www.sciencedirect.com/science/article/pii/S0165168402002505

    work page doi:10.1016/s0165-1684(02)00250-5 2002

  29. [29]

    Erdogan, A simple geometric blind source separation method for bounded mag- nitude sources, IEEE Transactions on Signal Processing 54 (2) (2006) 438–449

    A. Erdogan, A simple geometric blind source separation method for bounded mag- nitude sources, IEEE Transactions on Signal Processing 54 (2) (2006) 438–449. doi:10.1109/TSP.2005.861800

    work page doi:10.1109/tsp.2005.861800 2006

  30. [30]

    Vrins, J

    F. Vrins, J. A. Lee, M. Verleysen, A minimum-range approach to blind extraction of bounded sources, IEEE Transactions on Neural Networks 18 (3) (2007) 809–822. doi:10.1109/TNN.2006.889941

    work page doi:10.1109/tnn.2006.889941 2007

  31. [31]

    Tatli, A

    G. Tatli, A. T. Erdogan, Polytopic matrix factorization: Determinant maximization based criterion and identifiability, IEEE Transactions on Signal Processing 69 (2021) 5431–5447. doi:10.1109/TSP.2021.3112918

    work page doi:10.1109/tsp.2021.3112918 2021

  32. [32]

    Benign Overfitting in Binary Classification of Gaussian Mix- tures

    G. Tatli, A. T. Erdogan, Generalized polytopic matrix factorization, in: ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2021, pp. 3235–3239. doi:10.1109/ICASSP39728.2021.9413709. 23

    work page doi:10.1109/icassp39728.2021.9413709 2021

  33. [33]

    Brotto, K

    R. Brotto, K. Nose Filho, J. M. T. Romano, Alternative criteria for predictive blind deconvolution, Journal of Communication and Information Systems 33 (1) (2018)

    work page 2018

  34. [34]

    R. D. B. Brotto, K. Nose-Filho, J. M. T. Romano, Antisparse blind source separation, in: Proc. Signal Processing with Adaptive Sparse Structured Representations (SPARS) workshop, Vol. 52, 2019

    work page 2019

  35. [35]

    G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd Edition, John Hopkins, 1996

    work page 1996

  36. [36]

    S. A. W. Shah, K. Abed-Meraim, T. Y. Al-Naffouri, Blind source separation algorithms using hyperbolic and givens rotations for high-order qam constellations, IEEE Transac- tions on Signal Processing 66 (7) (2018) 1802–1816

    work page 2018

  37. [37]

    J. A. T. Thomas. M. Cover, Elements of Information Theory, Wiley-Interscience, 2006

    work page 2006

  38. [38]

    Tatli, A

    G. Tatli, A. T. Erdogan, Polytopic matrix factorization: Determinant maximization based criterion and identifiability, IEEE Transactions on Signal Processing 69 (2021) 5431–5447

    work page 2021

  39. [39]

    A. T. Erdogan, An information maximization based blind source separation approach for dependent and independent sources, in: ICASSP 2022-2022 IEEE International Confer- ence on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2022, pp. 4378–4382

    work page 2022

  40. [40]

    Tatli, A

    G. Tatli, A. T. Erdogan, A bayesian perspective for determinant minimization based robust structured matrix factorization, in: ICASSP 2023-2023 IEEE International Con- ference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2023, pp. 1–5

    work page 2023

  41. [41]

    Proakis, M

    J. Proakis, M. Salehi, Digital Communications, McGraw-Hill Education, 5th ed., 2007

    work page 2007

  42. [42]

    Jassim, S., Holubar, M., Richter, A., Wolff, C., Ohmer, X., and Bruni, E

    Q. Huynh-Thu, M. Ghanbari, Scope of validity of psnr in image/video quality assess- ment, Electronics Letters 44 (13) (2008) 800–801. doi:10.1049/el:20080522

    work page doi:10.1049/el:20080522 2008

  43. [43]

    Demarta, A

    S. Demarta, A. J. McNeil, The t copula and related copulas., International Statistical Review / Revue Internationale De Statistique 73 (1) (2005) 111–129. 24

    work page 2005

  44. [44]

    R. A. Horn, C. R. Johnson, Matrix Analysis, 2nd Edition, Cambridge University Press, 2012

    work page 2012

  45. [45]

    Foucart, H

    S. Foucart, H. Rauhut, A Mathematical Introduction to Compressive Sensing, 1st Edi- tion, Springer, 2013. A Proof Relying on the observation that the linear mixing process (invertible) ofNbounded sources, with finite real valued amplitudes in the interval [−A, A] will lead toNmixtures withℓ ∞- norm equal or bigger thanA, we show, inTheorem 1, that the equ...

    work page 2013

  46. [46]

    However, it is important to note that the condition g1 +g 2 +g 3 = 1 came from the time instantsn 0 andn 1, where the 3 sources assumed the values (A, A, A) and (−A,−A,−A)

    is a solution. However, it is important to note that the condition g1 +g 2 +g 3 = 1 came from the time instantsn 0 andn 1, where the 3 sources assumed the values (A, A, A) and (−A,−A,−A). These two conditions are not sufficient to guarantee the Extreme Points Condition. With such conditions, we must consider the time instants where the sources assume the ...

    work page

  47. [47]

    , we have y(n2) = A 3 + 2A 3 + 2A 3 = 5 A 3 > A and theℓ ∞-norm ofy(n) is not equal toA, which implies that ( −1 3 , 2 3 , 2

    work page

  48. [48]

    is no longer a solution. Hence, the Extreme Points Condition adds constraints to the vectorgthat exclude the possibility of spurious solutions,i.e., solutions that equalize theℓ ∞-norm but do not corre- spond to the canonical vectors. We can extend the result obtained for the extraction of one source to the separation of multiple ones. To do so, the mixin...

    work page