Topological data analysis "hearing" the shapes of drums and bells
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Topological data analysis “hearing” the shapes of drums and bells
By Guo-Wei Wei
January 13, 2023
arXiv:2301.05025v1 [math.HO] 11 Jan 2023
Abstract
Mark Kac asked a famous question in 1966, “Can one hear the shape of a drum?”, a spectral geometry
problem that has intrigued mathematicians for the last six decades and is important to many other fields,
such as architectural acoustics, audio forensics, pattern recognition, radiology, and imaging science. A
related question is how to hear the shape of a drum. We show that the answer was given in the set of 65
Zenghouyi chime bells dated back to 475-433 B.C. in China. The set of chime bells gradually varies their
sizes and weights to enable melodies, intervals, and temperaments. The same design principle was used
in many other musical instruments, such as xylophones, pan flutes, pianos, etc. We reveal that there
is a fascinating connection between the progression pattern of many musical instruments and filtration
(or spectral sequence) in topological data analysis (TDA). We argue that filtration-induced evolutionary
de Rham-Hodge theory provides a new mathematical foundation for musical instruments. Its discrete
counterpart, persistent Laplacians and many other persistent topological Laplacians, including persistent
sheaf Laplacians and persistent path Laplacians are briefly discussed.
“Can one hear the shape of a drum?”, a famous ques-
(a)
tion posed by Mark Kac in 1966 [10], has intrigued math-
ematicians for generations. In other words, if you hear
the sound from a drum, i.e., the set of overtones pro-
duced by the drum, can you infer its shape? Mathe-
matically, the essence of the spectral geometry question
is whether the shape can be uniquely determined from
the eigenvalues of the Laplacian operator defined on the
shape. There are many examples of isospectral mani- (b) (c) (d)
folds which are not isometric in two and higher dimen-
sional settings [6, 15, 17]. However, the problem is not
closed yet — one can discern the shapes of certain ge-
ometric types from their sounds [9, 16]. The question
has a far-reaching impact on many fields beyond math-
ematics, such as architectural acoustics, audio forensics,
pattern recognition, radiology, imaging science, and mu- Figure 1: The progression patterns in musical instruments.
sical science [3, 4]. a. The Zonghouyi chime bells. b. An xylophone. c. A
pan flute. d. Strings of a grand piano. Image courtesy of
In musical composition, it is a common practice to Wikipedia.
use a drum set of varying sizes, instead of a single piece
of drum, to facilitate a tonic harmonic progression, which is a foundation of harmony in modern music. The
human brain is trained from the drum set to distinguish the overtones of individual drum. The comparative
training/learning from a set of drums is the basis to hear the shape of a drum from a drum set.
Interestingly, to create tonal harmony, the ancient Chinese built a set of 65 Zenghouyi chime bells dated
back to 475-433 B.C. in the Warring States Period in China (Figure 1a). The shape of each chime bell was
designed to produce a distinct sound. The set of 65 chime bells gradually varies in their sizes and weights,
1
ranging from 153.4 centimeters (60.4 in) to 20.4 centimeters (8.0 in) in height and from 203.6 kilograms (449
lb) to 2.4 kilograms (5.3 lb) in weight. The set of Zenghouyi chime bells covers from C2 to D7 tonal range
and can play all twelve half tones in the middle area of the tonal range [21], enabling melodies, intervals, and
temperaments. The same design principle is used in many other musical instruments, such as xylophones
(Figure 1b), pan flutes (Figure 1c), and pianos (Figure 1d). Due to the close proximity among chime bells,
resonance among them may occur when they are struck, giving rise to prolonged harmony. Made of bronze,
one of the most precious metals available at the time, chime bells were used in various rituals, ceremonies,
and entertainment.
There is a fascinating and apparent connection between the progression pattern of musical instruments
(Figure 1) and mathematical filtration (Figure 2a), a spectral sequence technique [13] widely used in homo-
logical algebra and topological data analysis (TDA). As a new branch of mathematics, TDA uses topological
and geometric concepts to understand and extract topological patterns and structures in data. The main
workhorse of TDA is persistent homology [5, 22], a branch of algebraic topology that creates a mathematical
microscopy of a point cloud by filtration. Through the comparative analysis of the topological invariant
changes induced by the filtration, persistent homology delineates the shape of data [12]. Paired with ad-
vanced machine learning and deep learning algorithms, persistent homology has had tremendous success in
data science. It has been established as one of the most powerful tools in simplifying the geometric com-
plexity and reducing the high dimensionality of biomolecular interactions [11], revolutionizing drug discovery
and the forecasting of emerging viral variants.
However, persistent homology is not directly appli- (a)
cable to the tonal analysis of chime bells. First, the set a
of chime bells may be regarded as the result of a set of
evolving chime bell manifolds, rather than that of the fil-
tration of a point cloud. Additionally, there is no change (b)
in topological invariants associated the homotopic shape
evolution of the set of chime bells. Finally, persistent ho-
mology cannot present a frequency or tonal analysis of
chime bells or many other musical instruments.
Recently, an evolutionary de Rham-Hodge method (c)
has been proposed as a multiscale generalization of the
classical de Rham-Hodge theory, a landmark of the 20th
Century’s mathematics [1]. It provides a multiscale
geometric and topological analysis of filtration-induced Figure 2: Illustration of filtration and persistent Laplacian
manifolds, on which a family of evolutionary Hodge spectra. a. Filtration of a point cloud. b-c. The persistent
Laplacians can be defined on the set of chime bells to Laplacian analysis of the point cloud. Here, βjα and λαj (j =
characterize their tonal evolution. In association with 0, 1) are persistent Betti-j numbers and the first nonzero
eigenvalues of the jth persistent Laplacian, respectively.
a family of evolutionary de Rham complexes, evolution- Note that λα j capture the homotopic shape evolution of
ary Hodge Laplacians reveal the full set of topological data (see the frequency changes after α = 1.5) that is not
invariants, such as Betti numbers, in their kernel or null reflected in βjα . The similarity between λα α
0 and λ1 is a
space dimensions. coincidence. Image courtesy of Dr. Jian Jiang.
The point-cloud counterpart of the evolutionary de
Rham-Hodge method on manifolds is called persistent spectral graph [18] (also known as persistent Laplacians
[14]) on simplicial complexes. Like the evolutionary de Rham-Hodge method, persistent Laplacians not only
return the full set of topological invariants in their harmonic spectra as persistent homology does but also
capture the homotopic shape evolution of data during the filtration in their first non-harmonic spectra
(Figure 2b and 2c), for which persistent homology cannot describe.
A generalization of persistent Laplacians was made through the sheaf theory [8] and the resulting per-
2
sistent sheaf Laplacians [20] allow the embedding of heterogeneous characters in topological invariants,
e.g., encoding non-geometric information in a geometry-based simplicial complex. Another generalization
is persistent path Laplacians [19], built from the path complex and path homology [2, 7]. Persistent path
Laplacians are designed for directed graphs and directed networks. These new persistent topological Lapla-
cians not only lay a mathematical foundation for the tonal analysis in musical science but also significantly
extend the applicable domain and power of TDA.
References
[1] J. Chen, R. Zhao, Y. Tong, and G.-W. Wei. Evolutionary de Rham-Hodge method. Discrete and
Continuous Dynamical Systems. Series B, 26(7):3785, 2021.
[2] S. Chowdhury and F. Mémoli. Persistent path homology of directed networks. In Proceedings of the
Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1152–1169. SIAM, 2018.
[3] L. Cosmo, M. Panine, A. Rampini, M. Ovsjanikov, M. M. Bronstein, and E. Rodolà. Isospectralization,
or how to hear shape, style, and correspondence. In Proceedings of the IEEE/CVF Conference on
Computer Vision and Pattern Recognition, pages 7529–7538, 2019.
[4] I. Dokmanić, R. Parhizkar, A. Walther, Y. M. Lu, and M. Vetterli. Acoustic echoes reveal room shape.
Proceedings of the National Academy of Sciences, 110(30):12186–12191, 2013.
[5] H. Edelsbrunner, J. Harer, et al. Persistent homology-A survey. Contemporary Mathematics, 453:257–
282, 2008.
[6] C. Gordon, D. Webb, and S. Wolpert. Isospectral plane domains and surfaces via riemannian orbifolds.
Inventiones Mathematicae, 110(1):1–22, 1992.
[7] A. Grigor’yan, Y. Lin, Y. Muranov, and S.-T. Yau. Homologies of path complexes and digraphs. arXiv
preprint arXiv:1207.2834, 2012.
[8] J. Hansen and R. Ghrist. Toward a spectral theory of cellular sheaves. Journal of Applied and Compu-
tational Topology, 3(4):315–358, 2019.
[9] H. Hezari, Z. Lu, and J. Rowlett. The dirichlet isospectral problem for trapezoids. Journal of Mathe-
matical Physics, 62(5):051511, 2021.
[10] M. Kac. Can one hear the shape of a drum? The American Mathematical Monthly, 73(4P2):1–23, 1966.
[11] J. Liu, K.-L. Xia, J. Wu, S. S.-T. Yau, and G.-W. Wei. Biomolecular topology: Modelling and analysis.
Acta Mathematica Sinica, English Series, 38(10):1901–1938, 2022.
[12] P. Y. Lum, G. Singh, A. Lehman, T. Ishkanov, M. Vejdemo-Johansson, M. Alagappan, J. Carlsson, and
G. Carlsson. Extracting insights from the shape of complex data using topology. Scientific Reports,
3(1):1–8, 2013.
[13] J. McCleary. A user’s guide to spectral sequences. Number 58. Cambridge University Press, 2001.
[14] F. Mémoli, Z. Wan, and Y. Wang. Persistent laplacians: Properties, algorithms and implications. SIAM
Journal on Mathematics of Data Science, 4(2):858–884, 2022.
[15] J. Milnor. Eigenvalues of the Laplace operator on certain manifolds. Proceedings of the National
Academy of Sciences, 51(4):542–542, 1964.
[16] A. W. Reid. Isospectrality and commensurability of arithmetic hyperbolic 2-and 3-manifolds. Duke
Mathematical Journal, 65(2):215–228, 1992.
3
[17] M.-F. Vignéras. Variétés riemanniennes isospectrales et non isométriques. Annals of Mathematics,
112(1):21–32, 1980.
[18] R. Wang, D. D. Nguyen, and G.-W. Wei. Persistent spectral graph. International Journal for Numerical
Methods in Biomedical Engineering, 36(9):e3376, 2020.
[19] R. Wang and G.-W. Wei. Persistent path Laplacian. Foundations of Data Sciences, 5(1):26–55, 2023.
[20] X. Wei and G.-W. Wei. Persistent sheaf Laplacians. arXiv preprint arXiv:2112.10906, 2021.
[21] Y. Yan, K. Chai, H. Liang, and L. Kong. Physics involvement in ancient chinese chime bells. In AIP
Conference Proceedings, volume 1517, pages 43–48. American Institute of Physics, 2013.
[22] A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput. Geom., 33:249–274,
2005.
Guo-Wei Wei is an MSU Foundation Professor at Michigan State University. His research concerns the
mathematical foundations of biological science and data science.
4
You can also read
