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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.
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Capacity of Multiple One-Bit Transceivers in a Rayleigh Environment
We analyze the channel capacity of a system with a large number of one-bit transceivers in a classical Rayleigh environment with perfect channel information at the receiver. With M transmitters and N =alpha*M receivers, we derive an expression of the capacity per transmitter C, where C <= min(1; aalpha), as a function of alpha and signal-to-noise ratio (SNR) rho, when M -> infinity. We show that our expression is a good approximation for small M, and provide simple approximations of C for various ranges of alpha and rho. We conclude that at high SNR, C reaches its upper limit of one only if alpha > 1:24. Expressions for determining when C “saturates” as a function of alpha and rho are given.
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- Award ID(s):
- 1731056
- PAR ID:
- 10061921
- Date Published:
- Journal Name:
- IEEE Wireless Communications and Networking Conference
- ISSN:
- 1525-3511
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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