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Register a new userUpper and Lower Bounds on the Performance of Kernel PCA
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Benjamin Guedj
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Principal Component Analysis (PCA) is a popular method for dimension reduction and has attracted an unfailing interest for decades. Recently, kernel PCA has emerged as an extension of PCA but, despite its use in practice, a sound theoretical understanding of kernel PCA is missing. In this paper, we contribute lower and upper bounds on the efficiency of kernel PCA, involving the empirical eigenvalues of the kernel Gram matrix. Two bounds are for fixed estimators, and two are for randomized estimators through the PAC-Bayes theory. We control how much information is captured by kernel PCA on average, and we dissect the bounds to highlight strengths and limitations of the kernel PCA algorithm. Therefore, we contribute to the better understanding of kernel PCA. Our bounds are briefly illustrated on a toy numerical example.
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The principle of optimism in the face of uncertainty is one of the most widely used and successful ideas in multi-armed bandits and reinforcement learning. However, existing optimistic algorithms (primarily UCB and its variants) are often unable to deal with large context spaces. Essentially all existing well performing algorithms for general contextual bandit problems rely on weighted action allocation schemes; and theoretical guarantees for optimism-based algorithms are only known for restricted formulations. In this paper we study general contextual bandits under the realizability condition, and propose a simple generic principle to design optimistic algorithms, dubbed Upper Counterfactual Confidence Bounds (UCCB). We show that these algorithms are provably optimal and efficient in the presence of large context spaces. Key components of UCCB include: 1) a systematic analysis of confidence bounds in policy space rather than in action space; and 2) the potential function perspective that is used to express the power of optimism in the contextual setting. We further show how the UCCB principle can be extended to infinite action spaces, by constructing confidence bounds via the newly introduced notion of counterfactual action divergence.
We derive upper and lower bounds on the fidelity susceptibility in terms of macroscopic thermodynamical quantities, like susceptibilities and thermal average values. The quality of the bounds is checked by the exact expressions for a single spin in an external magnetic field. Their usefulness is illustrated by two examples of many-particle models which are exactly solved in the thermodynamic limit: the Dicke superradiance model and the single impurity Kondo model. It is shown that as far as divergent behavior is considered, the fidelity susceptibility and the thermodynamic susceptibility are equivalent for a large class of models exhibiting critical behavior.
Positive semidefinite rank (PSD-rank) is a relatively new quantity with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix $M$ as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.
The emergence of systems with non-volatile main memory (NVM) increases the interest in the design of emph{recoverable concurrent objects} that are robust to crash-failures, since their operations are able to recover from such failures by using state retained in NVM. Of particular interest are recoverable algorithms that, in addition to ensuring object consistency, also provide emph{detectability}, a correctness condition requiring that the recovery code can infer if the failed operation was linearized or not and, in the former case, obtain its response. In this work, we investigate the space complexity of detectable algorithms and the external support they require. We make the following three contributions. First, we present the first wait-free bounded-space detectable read/write and CAS object implementations. Second, we prove that the bit complexity of every $N$-process obstruction-free detectable CAS implementation, assuming values from a domain of size at least $N$, is $Omega(N)$. Finally, we prove that the following holds for obstruction-free detectable implementations of a large class of objects: their recoverable operations must be provided with emph{auxiliary state} -- state that is not required by the non-recoverable counterpart implementation -- whose value must be provided from outside the operation, either by the system or by the caller of the operation. In contrast, this external support is, in general, not required if the recoverable algorithm is not detectable.
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds are optimal for absolutely monotone potentials in the sense that for the linear programming technique they cannot be improved by using polynomials of the same or lower degree. When additional information about the structure (upper and lower bounds for the inner products) of the designs is known, improvements on the bounds are obtained. Furthermore, we provide `test functions for determining when the linear programming lower bounds for energy can be improved utilizing higher degree polynomials. We also provide some asymptotic results for these energy bounds.
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