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![Leverage
Optimization
(v1)
Using
proximal
linearized
minimization
[1],
the
update
rule
for
solving
above
optimization
is
[1] J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,”
Mathematical Programming, vol. 146, no. 1-2, pp. 459–494, 2014.
where
the
proximal
mapping
becomes
soft-‐thresholding:
where
𝛾̅ = 𝛾 − 1.](https://image.slidesharecdn.com/kernelrkhsandgp-170328222447/85/Kernel-RKHS-and-Gaussian-Processes-39-320.jpg)
Uploaded bySungjoon Choi
Kernel, RKHS, and Gaussian Processes
The document discusses leveraged Gaussian process regression, emphasizing its ability to anchor positive training data while avoiding negative data. It introduces a sparse constrained leverage optimization approach to handle the challenges associated with the number of leverage parameters, proposing new methods for more accurate optimization. Additionally, it outlines applications in sensory field reconstruction and autonomous driving experiments, highlighting the advantages of these techniques.
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Kernel, RKHS, and Gaussian Processes
- 1. Kernel, RKHS, and GaussianProcesses Caution! No proof will be given. Sungjoon Choi, SNU
- 31.
- 32. Leveraged Gaussian Processes The original Gaussian process regression anchors positive training data. The proposed leveraged Gaussian process regression anchors positive data while avoiding negative data. Moreover, it is possible to vary the leverage of each training data from -‐1 to +1.
- 33. Positive and NegativeMotion Control Data for Autonomous Navigation
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- 38. Leverage Optimization • The key intuition behind the leverage optimization is that we cast the leverage optimization problem into a model selection problem in Gaussian process regression. • However, the number of leverage parameters is equivalent to the number of training data. • To handle this issue, we propose a sparse constrained leverage optimization where we assume that the majority of leverage parameters are +1.
- 39. Leverage Optimization (v1) Using proximal linearized minimization [1], the update rule for solving above optimization is [1] J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Mathematical Programming, vol. 146, no. 1-2, pp. 459–494, 2014. where the proximal mapping becomes soft-‐thresholding: where 𝛾̅ = 𝛾 − 1.
- 40. Leverage Optimization (v2) We propose a new leverage optimization method by doubling the leverage parameters to positive and negative parts. Furthermore, we assume multiple demonstrations are collected from one demonstrator. By doing so, we have two major benefits: 1. As the L1-‐norm regularizer is only on the positive parts of the leverages, negative parts of the leverages can be optimized more accurately. 2. By doubling the variables, proximal mapping is no longer needed.
- 41. Sensory Field Reconstruction Sensory observations from correlated sensory fields Leverage Optimization Collect observations
- 42. Sensory observations from correlated sensory fields Proposed Method Previous Method Without Optimization Sensory Field Reconstruction
- 43. Autonomous Driving Experiments Driving demonstrations with mixed qualities Leverage Optimization Collect observations
- 44. Proposed Method Previous Method Autonomous Driving Experiments
- 45. Proposed MethodWithout Optimization Autonomous Driving Experiments