Learning with Fredholm Kernels
In this paper we propose a framework for supervised and semi-supervised learning based on reformulating the learning problem as a regularized Fredholm integral equation. Our approach fits naturally into the kernel framework and can be interpreted as constructing new data-dependent kernels, which we call Fredholm kernels. We proceed to discuss the "noise assumption" for semi-supervised learning and provide both theoretical and experimental evidence that Fredholm kernels can effectively utilize unlabeled data under the noise assumption. We demonstrate that methods based on Fredholm learning show very competitive performance in the standard semi-supervised learning setting.
This paper describes a framework particularly useful for semi-supervised learning based on Fredholm kernels.  The classical supervised learning optimization problem solved in kernel-based methods is extended to incorporate unlabeled information leading to discretized version of the Fredholm integral equation.QualityThe paper has high technical quality with well-supported claims by theoretical analysis and convincing experimental results.  The proposed formulation leads to a new data-dependent kernel that incorporates unlabeled information.  The classifying function differs from the classical represent theorem solution but it is still elegant and easy to compute.  I have few comments: One is  related to Equation 3 where the solution of the proposed optimization problem is described.  Since the Authors mentioned that the associated kernel need not be PSD, does Equation 3 still hold in this case ? A proof on how Equation 3 was obtained could have been nice to see too.  Another comment concerns the experimental section where the Authors chose an "optimal" parameter for all methods in the benchmark.  How were those parameters determined ? What is meant with "optimal" ?ClarityThe paper is clearly written and well-organized in most part.  It is easy to follow and the main ideas are explain adequately.  There are some typos though for example:* Line 82:  "... will a Euclidean ..." -> ... will be a Euclidean ..."* Line 83: "or" missing.* Section 4 title: "Fredhom" -> Fredholm* Line 247: "porportional" -> proportional* Line 283: "That is, or..." -> That is, for...* Line 352: Sentence needs to be rewritten.OriginalityThe proposed approach is related to a recent NIPS publication ([9]) but it is novel in essence.  It is clear how the work differs from this publication and others.  SignificanceThe results are important and significant.  There was an significant effort to test the proposed method on several datasets in different application domains.