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arxiv: 2605.17117 · v1 · pith:I6BTU2F3new · submitted 2026-05-16 · 💱 q-fin.ST

Geometric Observables for Financial Regime Detection

Will Hammond This is my paper

Pith reviewed 2026-05-20 15:08 UTC · model grok-4.3

classification 💱 q-fin.ST
keywords geometric observablesfinancial regime detectionspectral embeddingberry phase rateregime shiftsequity index returnsunsupervised detectioncrisis monitoring
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The pith

Geometric observables from a spectral embedding of equity returns detect market regime shifts with a median Cohen's d of 0.72 and fewer false alarms than supervised random forests.

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

The paper establishes that four quantities extracted from the geometry of a learned spectral embedding of stock-index returns can serve as unsupervised detectors of financial regime shifts across 17 crises from 2000 to 2024. A reader would care because accurate regime detection supports better risk management and allocation decisions, and these signals require little labeled crisis data beyond hyperparameter tuning. The Berry Phase Rate in particular shows an out-of-sample effect size of 0.72 while cutting false alarms by roughly two-thirds relative to a label-supervised random forest. Geometric and classical indicators remain largely uncorrelated, indicating they track separate aspects of market behavior. If the geometric channel proves stable, it offers a complementary, label-light route to monitoring systemic risk that does not depend on historical crisis annotations for core scoring.

Core claim

From a learned spectral embedding of equity-index returns the authors extract Berry Phase Rate, Spectral Entropy, Reduced State Purity, and Hamiltonian Sensitivity. These four geometric observables function as regime-shift detectors. Evaluated walk-forward against 46 baselines on 17 historical crises, the Berry Phase Rate records a median out-of-sample Cohen's d of 0.72 with a 95 percent bootstrap interval of 0.34 to 1.18 and yields approximately 1.2 false alarms per year versus 3.6 for a supervised random forest. Reduced State Purity reaches the highest in-sample separability of any method tested, while the geometric and classical families show mean absolute correlation near 0.22. Score use

What carries the argument

The learned spectral embedding of equity-index returns, from which the four geometric observables are computed to quantify changes in the underlying market geometry.

If this is right

  • The Berry Phase Rate could be integrated into existing risk systems to lower the rate of spurious regime alerts by roughly two-thirds.
  • Because geometric and classical observables are largely uncorrelated, combining the two families may improve overall detection accuracy without increasing model complexity.
  • Reduced State Purity's strong in-sample separability suggests it could be used to rank the intensity of different regime transitions.
  • The unsupervised construction of the scores means the approach remains usable even when labeled crisis examples are scarce.

Where Pith is reading between the lines

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

  • One could examine whether the same observables remain informative when applied to individual stocks or to intraday data rather than daily index returns.
  • The low correlation with classical methods raises the possibility that geometric signals could flag regime changes driven by microstructure or liquidity effects that standard volatility measures miss.
  • If the embedding step is robust, extending the observables to multi-market or cross-asset embeddings might reveal synchronized regime shifts across global equity indices.

Load-bearing premise

The spectral embedding must preserve geometric properties that correspond to genuine economic regime shifts rather than being driven primarily by preprocessing choices or embedding artifacts.

What would settle it

A new out-of-sample period after 2024 in which the Berry Phase Rate shows no statistically significant separation around known market events or produces a higher annual false-alarm rate than the random-forest baseline.

Figures

Figures reproduced from arXiv: 2605.17117 by Will Hammond.

Figure 1
Figure 1. Figure 1: Quantum metric eigenvalue spectra at normal vs. crisis timepoints. (a) Mean spectrum with [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geometric crisis anatomy: 2008 GFC. Eight panels show SPY price, daily returns, the four geometric observables [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Geometric crisis anatomy: 2020 COVID-19. [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Geometric crisis anatomy: 2022 rate hikes. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Cohen’s d distributions across 17 crises. Dashed line: d = 0.8 (large effect). 2008 2012 2016 2020 2024 Date 3.0 3.5 4.0 4.5 5.0 Spectral Gap (x) Asian Crisis 1997 LTCM/Russia 1998 Dot-Com Crash 2000 September 11 2001 Quant Crisis 2007 GFC 2008 Flash Crash 2010 Euro Crisis 2011 Taper Tantrum 2013 China Crash 2015 Brexit 2016 Volmageddon 2018 Q4 Selloff 2018 Repo Crisis 2019 COVID 2020 Meme/Archegos 2021 Ra… view at source ↗
Figure 6
Figure 6. Figure 6: Spectral gap dynamics across 4 crises. The gap remains strictly positive ( [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Curvature–gap bound verification. Berry curvature magnitude vs. [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Rolling Chern number distribution. Values cluster near integers during normal periods (21.3% within 0.1 of an [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: QFI–metric identity (Proposition 2). Finite-difference metric vs. perturbation-theory metric at 500 time steps. [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
read the original abstract

We extract four geometric observables -- Berry Phase Rate, Spectral Entropy, Reduced State Purity, and Hamiltonian Sensitivity -- from a learned spectral embedding of equity-index returns and evaluate them as regime-shift detectors against 46 classical and machine-learning baselines on 17 historical crises spanning 2000-2024. Under walk-forward nested hyperparameter selection on nine labelled windows, the Berry Phase Rate achieves an unbiased out-of-sample median Cohen's $d = 0.72$ (95% percentile-bootstrap CI $[0.34, 1.18]$, 10,000 resamples) and produces approximately 67% fewer false alarms per year than a label-supervised Random Forest (1.2 vs. 3.6 per year). Reduced State Purity attains the highest in-sample separability of any method ($d = 0.83$), tied closely by the Absorption Ratio ($d = 0.80$); geometric and classical channels are largely uncorrelated (mean $|\rho| \approx 0.22$), suggesting they capture distinct risk signals. Score construction is unsupervised; hyperparameter selection is the only supervised step.

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

1 major / 2 minor

Summary. The manuscript extracts four geometric observables—Berry Phase Rate, Spectral Entropy, Reduced State Purity, and Hamiltonian Sensitivity—from a learned spectral embedding of equity-index returns. These are evaluated as regime-shift detectors against 46 classical and machine-learning baselines across 17 historical crises (2000–2024). Under walk-forward nested hyperparameter selection on nine labelled windows, the Berry Phase Rate is reported to achieve an out-of-sample median Cohen’s d = 0.72 (95% percentile-bootstrap CI [0.34, 1.18]) and approximately 67% fewer false alarms per year than a label-supervised Random Forest, while geometric and classical channels remain largely uncorrelated (mean |ρ| ≈ 0.22).

Significance. If the performance claims hold after addressing potential label leakage in embedding construction, the work supplies new, largely orthogonal geometric signals for financial regime detection. The reported bootstrap confidence intervals, walk-forward validation, and explicit separation of unsupervised score construction from the single supervised hyperparameter step are methodological strengths that support empirical credibility.

major comments (1)
  1. [§4] §4 (Methods, hyperparameter selection): The walk-forward nested selection of spectral embedding hyperparameters (kernel scale, embedding dimension, affinity construction) is performed on the nine labelled windows that overlap with the crises used for evaluation. This procedure risks implicit optimization of the embedding for separability on known regime shifts, which would undermine the claim that the Berry Phase Rate observables are unbiased out-of-sample detectors. A concrete description of the nested cross-validation folds and confirmation that no evaluation-window labels influence the chosen hyperparameters is required to support the central performance claims.
minor comments (2)
  1. [Abstract] Abstract and §3: The statement that 'Score construction is unsupervised; hyperparameter selection is the only supervised step' is helpful but should be repeated with a short diagram or pseudocode in the main text to clarify the information flow for readers concerned about circularity.
  2. [Results] Table 2 or equivalent results table: The false-alarm rates (1.2 vs. 3.6 per year) would benefit from an explicit definition of what constitutes a 'false alarm' (e.g., threshold, persistence window) to allow direct replication.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and positive assessment of the methodological strengths, including the walk-forward validation and separation of unsupervised score construction from the supervised hyperparameter step. We address the major comment on hyperparameter selection below and will revise the manuscript accordingly to provide the requested clarifications.

read point-by-point responses

  1. Referee: [§4] §4 (Methods, hyperparameter selection): The walk-forward nested selection of spectral embedding hyperparameters (kernel scale, embedding dimension, affinity construction) is performed on the nine labelled windows that overlap with the crises used for evaluation. This procedure risks implicit optimization of the embedding for separability on known regime shifts, which would undermine the claim that the Berry Phase Rate observables are unbiased out-of-sample detectors. A concrete description of the nested cross-validation folds and confirmation that no evaluation-window labels influence the chosen hyperparameters is required to support the central performance claims.

    Authors: We agree that a more explicit description of the nested cross-validation procedure is necessary to fully address concerns about potential label leakage. In the current manuscript, the nine labelled windows are used exclusively for hyperparameter selection via walk-forward nested CV: the inner loop selects kernel scale, embedding dimension, and affinity parameters by maximizing a separability metric computed only on training folds within those windows, while the outer loop validates on subsequent held-out segments of the same nine windows. The selected hyperparameters are then fixed and applied to compute observables on the full set of 17 crises for out-of-sample evaluation. No labels from the eight additional evaluation crises (or from test segments within the nine windows) are used during selection. To strengthen this, we will add a detailed schematic of the nested folds, explicit pseudocode, and a statement confirming that evaluation-window labels exert no influence on the chosen hyperparameters. This revision will be incorporated in the next version. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation of geometric observables

full rationale

The paper's central derivation extracts Berry Phase Rate, Spectral Entropy, Reduced State Purity, and Hamiltonian Sensitivity directly from a spectral embedding of returns; these quantities are computed without labels in their definitions. Hyperparameter selection for the embedding occurs via walk-forward nested selection on nine labelled windows, but the abstract explicitly isolates this as the sole supervised step while declaring score construction unsupervised and reporting unbiased out-of-sample metrics on the remaining crises. The reported median Cohen's d = 0.72 and false-alarm reduction are measured on held-out data, and the paper notes low correlation (mean |ρ| ≈ 0.22) with classical baselines, confirming the observables carry independent content rather than reducing to the labelled windows by construction. No equation or step equates the final performance figures or observables to the input labels or selection objective.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the spectral embedding as a faithful geometric representation of financial returns and on the accuracy of the historical crisis labels used for both hyperparameter tuning and performance measurement.

free parameters (1)
  • Spectral embedding hyperparameters
    Chosen via supervised walk-forward nested selection on nine labelled windows; these control the embedding used to compute all four geometric observables.
axioms (1)
  • domain assumption The learned spectral embedding of equity-index returns captures geometric properties relevant to regime shifts.
    Invoked when defining and extracting Berry Phase Rate, Spectral Entropy, Reduced State Purity, and Hamiltonian Sensitivity from the embedding.

pith-pipeline@v0.9.0 · 5709 in / 1378 out tokens · 84328 ms · 2026-05-20T15:08:16.265307+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/ArrowOfTime.lean forward_accumulates, arrow_from_z echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Berry Phase Rate is the absolute increment of Berry curvature between consecutive time steps: ˙γ(t) = |F01(xt)−F01(xt−1)| (Eq. 5), computed via the discrete plaquette method.

  • IndisputableMonolith/Foundation/ArrowOfTime.lean z_monotone_absolute echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Theorem 1 (Smoothness): positive spectral gap Δ(x) > 0 guarantees C^∞ geometry (metric, curvature). Empirical validation shows gap remains positive and opens during crises.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 2 internal anchors

  1. [1]

    Kritzman, Y

    M. Kritzman, Y. Li, S. Page, R. Rigobon, Principal components as a measure of systemic risk, The Journal of Portfolio Management 37 (4) (2011) 112–126. 17 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 1/∆2 0.2 0.4 0.6 0.8 1.0 ∥Fab∥F (Berry curvature) Curvature-Gap Relationship (Theorem 3) Normal Crisis Theorem 3 bound (C=5.83e+00) Figure 7: Curvature–gap b...

    work page 2011

  2. [2]

    J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 (2) (1989) 357–384.doi:10.2307/1912559

    work page doi:10.2307/1912559 1989

  3. [3]

    A. Ang, G. Bekaert, International asset allocation with regime shifts, Review of Financial Studies 15 (4) (2002) 1137–1187.doi:10.1093/rfs/15.4.1137

    work page doi:10.1093/rfs/15.4.1137 2002

  4. [4]

    R. P. Adams, D. J. C. MacKay, Bayesian online changepoint detection, arXiv preprint arXiv:0710.3742 (2007)

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  5. [5]

    Candelori, A

    L. Candelori, A. G. Abanov, J. Berger, C. J. Hogan, V. Kirakosyan, K. Musaelian, R. Samson, J. E. T. Smith, D. Villani, M. T. Wells, M. Xu, Robust estimation of the intrinsic dimension of data sets with quantum cognition machine learning, Scientific Reports 15 (2025) 6933.doi: 10.1038/s41598-025-91676-8

    work page doi:10.1038/s41598-025-91676-8 2025

  6. [6]

    A. G. Abanov, L. Candelori, H. C. Steinacker, M. T. Wells, J. R. Busemeyer, V. Kirakosyan, N. Marzari, S. Pinnamaneni, D. Villani, M. Xu, K. Musaelian, Quantum geometry of data, arXiv preprint arXiv:2507.21135 (2025)

    work page arXiv 2025

  7. [7]

    Timmermann, Forecast combinations, in: G

    A. Timmermann, Forecast combinations, in: G. Elliott, C. W. J. Granger, A. Timmermann (Eds.), Handbook of Economic Forecasting, Vol. 1, Elsevier, 2006, Ch. 4, pp. 135–196

    work page 2006

  8. [8]

    J. M. Bates, C. W. J. Granger, The combination of forecasts, Journal of the Operational Research Society 20 (4) (1969) 451–468

    work page 1969

  9. [9]

    Amari, H

    S. Amari, H. Nagaoka, Methods of Information Geometry, American Mathematical Society, 2000

    work page 2000

  10. [10]

    D. C. Brody, L. P. Hughston, Interest rates and information geometry, Proceedings of the Royal Society A 457 (2010) (2001) 1343–1363.doi:10.1098/rspa.2000.0722

    work page doi:10.1098/rspa.2000.0722 2010

  11. [11]

    Taylor, Clustering financial return distributions using the Fisher information metric, Entropy 21 (2) (2019) 110.doi:10.3390/e21020110

    S. Taylor, Clustering financial return distributions using the Fisher information metric, Entropy 21 (2) (2019) 110.doi:10.3390/e21020110

    work page doi:10.3390/e21020110 2019

  12. [12]

    Horvath, Z

    B. Horvath, Z. Issa, A. Muguruza, Clustering market regimes using the Wasserstein distance, Journal of Computational Finance (2024).doi:10.21314/jcf.2024.005

    work page doi:10.21314/jcf.2024.005 2024

  13. [13]

    R. S. Sandhu, T. T. Georgiou, E. Reznik, L. Zhu, I. Kolesov, Y. Senbabaoglu, A. Tannenbaum, Ricci curvature: An economic indicator for market fragility and systemic risk, Science Advances 2 (6) (2016) e1501495.doi:10.1126/sciadv.1501495. 19

    work page doi:10.1126/sciadv.1501495 2016

  14. [14]

    Gidea, Y

    M. Gidea, Y. A. Katz, Topological data analysis of financial time series: Landscapes of crashes, Physica A 491 (2018) 820–834.doi:10.1016/j.physa.2017.09.028

    work page doi:10.1016/j.physa.2017.09.028 2018

  15. [15]

    Guritanu, E

    E. Guritanu, E. Barbierato, A. Gatti, Topological machine learning for financial crisis detection: Early warning signals from persistent homology, Computers 14 (10) (2025) 408.doi:10.3390/ computers14100408

    work page 2025

  16. [16]

    Noise Dressing of Financial Correlation Matrices

    L. Laloux, P. Cizeau, J.-P. Bouchaud, M. Potters, Noise dressing of financial correlation matrices, Physical Review Letters 83 (7) (1999) 1467–1470.doi:10.1103/PhysRevLett.83.1467

    work page doi:10.1103/physrevlett.83.1467 1999

  17. [17]

    Z. Issa, B. Horvath, Non-parametric online market regime detection and regime clustering for multidi- mensional and path-dependent data structures, arXiv preprint arXiv:2306.15835 (2023)

    work page arXiv 2023

  18. [18]

    M. M. Bronstein, J. Bruna, T. Cohen, P. Veličković, Geometric deep learning: Grids, groups, graphs, geodesics, and gauges, arXiv preprint arXiv:2104.13478 (2021)

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  19. [19]

    Journal of Econo- metrics 31(3):307–327, DOI 10.1016/0304-4076(86)90063-1

    T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (3) (1986) 307–327.doi:10.1016/0304-4076(86)90063-1

    work page doi:10.1016/0304-4076(86)90063-1 1986

  20. [20]

    Musaelian, A

    K. Musaelian, A. Abanov, J. Berger, L. Candelori, V. Kirakosyan, R. Samson, J. Smith, D. Villani, Quantum cognition machine learning: AI needs quantum, Tech. rep., Qognitive, Inc., technical white paper (not peer-reviewed) (2024)

    work page 2024

  21. [21]

    Samson, J

    R. Samson, J. Berger, L. Candelori, V. Kirakosyan, K. Musaelian, D. Villani, Quantum cognition machine learning: Financial forecasting, Tech. rep., Qognitive, Inc., technical white paper (not peer- reviewed) (2024)

    work page 2024

  22. [22]

    J. P. Provost, G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Mathematical Physics 76 (3) (1980) 289–301.doi:10.1007/BF02193559

    work page doi:10.1007/bf02193559 1980

  23. [23]

    M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proceedings of the Royal Society A 392 (1802) (1984) 45–57.doi:10.1098/rspa.1984.0023

    work page doi:10.1098/rspa.1984.0023 1984

  24. [24]

    S. L. Braunstein, C. M. Caves, Statistical distance and the geometry of quantum states, Physical Review Letters 72 (22) (1994) 3439–3443.doi:10.1103/PhysRevLett.72.3439

    work page doi:10.1103/physrevlett.72.3439 1994

  25. [25]

    Electro nic transport in two-dimensional graphene

    L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Reviews of Modern Physics 90 (2018) 035005.doi:10.1103/RevModPhys. 90.035005

    work page doi:10.1103/revmodphys 2018

  26. [26]

    G. Tóth, I. Apellaniz, Quantum metrology from a quantum information science perspective, Journal of Physics A: Mathematical and Theoretical 47 (42) (2014) 424006.doi:10.1088/1751-8113/47/42/ 424006

    work page doi:10.1088/1751-8113/47/42/ 2014

  27. [27]

    Cramér, Mathematical Methods of Statistics, Princeton University Press, 1946

    H. Cramér, Mathematical Methods of Statistics, Princeton University Press, 1946

    work page 1946

  28. [28]

    C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, 1976

    work page 1976

  29. [29]

    Laloux, P

    L. Laloux, P. Cizeau, M. Potters, J.-P. Bouchaud, Random matrix theory and financial correla- tions, International Journal of Theoretical and Applied Finance 3 (3) (2000) 391–397.doi:10.1142/ S0219024900000255

    work page 2000

  30. [30]

    J. Bai, S. Ng, Determining the number of factors in approximate factor models, Econometrica 70 (1) (2002) 191–221.doi:10.1111/1468-0262.00273

    work page doi:10.1111/1468-0262.00273 2002

  31. [31]

    Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase, Physical Review Letters 51 (24) (1983) 2167–2170.doi:10.1103/PhysRevLett.51.2167

    B. Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase, Physical Review Letters 51 (24) (1983) 2167–2170.doi:10.1103/PhysRevLett.51.2167. 20

    work page doi:10.1103/physrevlett.51.2167 1983

  32. [32]

    G. Chow, E. Jacquier, M. Kritzman, K. Lowry, Optimal portfolios in good times and bad, Financial Analysts Journal 55 (3) (1999) 65–73.doi:10.2469/faj.v55.n3.2273

    work page doi:10.2469/faj.v55.n3.2273 1999

  33. [33]

    D. N. Politis, H. White, Automatic block-length selection for the dependent bootstrap, Econometric Reviews 23 (1) (2004) 53–70.doi:10.1081/ETC-120028836

    work page doi:10.1081/etc-120028836 2004

  34. [34]

    Pancharatnam, Generalized theory of interference, and its applications, Proceedings of the Indian Academy of Sciences, Section A 44 (1956) 247–262.doi:10.1007/BF03046050

    S. Pancharatnam, Generalized theory of interference, and its applications, Proceedings of the Indian Academy of Sciences, Section A 44 (1956) 247–262.doi:10.1007/BF03046050

    work page doi:10.1007/bf03046050 1956

  35. [35]

    Chern, Characteristic classes of Hermitian manifolds, Annals of Mathematics 47 (1) (1946) 85–121

    S.-S. Chern, Characteristic classes of Hermitian manifolds, Annals of Mathematics 47 (1) (1946) 85–121. doi:10.2307/1969037

    work page doi:10.2307/1969037 1946

  36. [36]

    Nakahara, Geometry, Topology and Physics, 2nd Edition, Taylor & Francis, 2003

    M. Nakahara, Geometry, Topology and Physics, 2nd Edition, Taylor & Francis, 2003

    work page 2003

  37. [37]

    & Killoran, N

    M. Schuld, N. Killoran, Quantum machine learning in feature Hilbert spaces, Physical Review Letters 122 (2019) 040504.doi:10.1103/PhysRevLett.122.040504

    work page doi:10.1103/physrevlett.122.040504 2019

  38. [38]

    J. R. Busemeyer, P. D. Bruza, Quantum Models of Cognition and Decision, Cambridge University Press, 2012

    work page 2012

  39. [39]

    Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966

    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966. Appendix A. Proofs of Formal Results Proof of Theorem 1.We apply the implicit function theorem to the eigenvalue equation with a gauge-fixing constraint. DefineΦ:U×C n ×R→C n ×Rby Φ(x, ψ, E) = H(x)ψ−Eψ,⟨ψ 0(x0)|ψ⟩ −1 , where the second component fixes the gauge by requiring⟨ψ0(x0)|ψ...

    work page 1966