An official website of the United States government
Summary The fused lasso, also known as total-variation denoising, is a locally adaptive function estimator over a regular grid of design points. In this article, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to a method for nonparametric regression. This approach, which we call the $$K$$-nearest-neighbours fused lasso, involves computing the $$K$$-nearest-neighbours graph of the design points and then performing the fused lasso over this graph. We show that this procedure has a number of theoretical advantages over competing methods: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the $$K$$-nearest-neighbours approach. In a simulation study and an application to flu data, we show that excellent results are obtained. For completeness, we also study an estimator that makes use of an $$\epsilon$$-graph rather than a $$K$$-nearest-neighbours graph and contrast it with the $$K$$-nearest-neighbours fused lasso.
more »
« less
Learning the Helix Topology of Musical Pitch
To explain the consonance of octaves, music psychologists represent pitch as a helix where azimuth and axial coordinate correspond to pitch class and pitch height respectively. This article addresses the problem of discovering this helical structure from unlabeled audio data. We measure Pearson correlations in the constant-Q transform (CQT) domain to build a K-nearest neighbor graph between frequency subbands. Then, we run the Isomap manifold learning algorithm to represent this graph in a three-dimensional space in which straight lines approximate graph geodesics. Experiments on isolated musical notes demonstrate that the resulting manifold resembles a helix which makes a full turn at every octave. A circular shape is also found in English speech, but not in urban noise. We discuss the impact of various design choices on the visualization: instrumentarium, loudness mapping function, and number of neighbors K.
more »
« less
- Award ID(s):
- 1633206
- PAR ID:
- 10301445
- Date Published:
- Journal Name:
- ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- Page Range / eLocation ID:
- 11 to 15
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Image retrieval relies heavily on the quality of the data modeling and the distance measurement in the feature space. Building on the concept of image manifold, we first propose to represent the feature space of images, learned via neural networks, as a graph. Neighborhoods in the feature space are now defined by the geodesic distance between images, represented as graph vertices or manifold samples. When limited images are available, this manifold is sparsely sampled, making the geodesic computation and the corresponding retrieval harder. To address this, we augment the manifold samples with geometrically aligned text, thereby using a plethora of sentences to teach us about images. In addition to extensive results on standard datasets illustrating the power of text to help in image retrieval, a new public dataset based on CLEVR is introduced to quantify the semantic similarity between visual data and text data. The experimental results show that the joint embedding manifold is a robust representation, allowing it to be a better basis to perform image retrieval given only an image and a textual instruction on the desired modifications over the image.more » « less
-
Abstract In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). Our filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as manifold analogues of various popular GNNs. We propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating an underlying manifold by a sparse graph. We then prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity, and we numerically demonstrate its effectiveness on real-world and synthetic data sets.more » « less
-
Abstract Kernelized Gram matrix $$W$$ constructed from data points $$\{x_i\}_{i=1}^N$$ as $$W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {\sigma ^2} ) $$ is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $$\sigma $$, and a common practice called self-tuned kernel adaptively sets a $$\sigma _i$$ at each point $$x_i$$ by the $$k$$-nearest neighbor (kNN) distance. When $$x_i$$s are sampled from a $$d$$-dimensional manifold embedded in a possibly high-dimensional space, unlike with fixed-bandwidth kernels, theoretical results of graph Laplacian convergence with self-tuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $$L_N$$ to manifold (weighted-)Laplacian for a new family of kNN self-tuned kernels $$W^{(\alpha )}_{ij} = k_0( \frac{ \| x_i - x_j \|^2}{ \epsilon \hat{\rho }(x_i) \hat{\rho }(x_j)})/\hat{\rho }(x_i)^\alpha \hat{\rho }(x_j)^\alpha $$, where $$\hat{\rho }$$ is the estimated bandwidth function by kNN and the limiting operator is also parametrized by $$\alpha $$. When $$\alpha = 1$$, the limiting operator is the weighted manifold Laplacian $$\varDelta _p$$. Specifically, we prove the point-wise convergence of $$L_N f $$ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C^0$ consistency for $$\hat{\rho }$$ which bounds the relative estimation error $$|\hat{\rho } - \bar{\rho }|/\bar{\rho }$$ uniformly with high probability, where $$\bar{\rho } = p^{-1/d}$$ and $$p$$ is the data density function. Our theoretical results reveal the advantage of the self-tuned kernel over the fixed-bandwidth kernel via smaller variance error in low-density regions. In the algorithm, no prior knowledge of $$d$$ or data density is needed. The theoretical results are supported by numerical experiments on simulated data and hand-written digit image data.more » « less
-
Abstract This study presents observations of a large pseudostreamer solar eruption and, in particular, the post-eruption relaxation phase, as captured by Metis, on board the Solar Orbiter, on 2022 October 12, during its perihelion passage. Utilizing total-brightness data, we observe the outward propagation of helical features up to 3 solar radii along a radial column that appears to correspond to the stalk of the pseudostreamer. The helical structures persisted for more than 3 hr following a jet-like coronal mass ejection associated with a polar crown prominence eruption. A notable trend is revealed: the inclination of these features decreases as their polar angle and height increase. Additionally, we measured their helix pitch. Despite the 2 minute time cadence limiting direct correspondence among filamentary structures in consecutive frames, we find that the Metis helical structure may be interpreted as a consequence of twist (nonlinear torsional Alfvén waves) and plasma liberated by interchange reconnection. A comparison was performed between the helix parameters as outlined by fine-scale outflow features and those obtained from synthetic white-light images derived from the high-resolution magnetohydrodynamics simulation of interchange reconnection in a pseudostreamer topology by P. F. Wyper et al. A remarkable similarity between the simulation-derived images and the observations was found. We conjecture that these Metis observations may represent the upper ends of the spatial and energy scales of the interchange reconnection process that has been proposed recently as the origin of the Alfvénic solar wind.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ICASSP40776.2020.9053644
