Non-linear Dimensionality Regularizer for Solving Inverse Problems
Consider an ill-posed inverse problem of estimating causal factors from observations, one of which is known to lie near some (unknown) low-dimensional, non-linear manifold expressed by a predefined Mercer-kernel. Solving this problem requires simultaneous estimation of these factors and learning the low-dimensional representation for them. In this work, we introduce a novel non-linear dimensionality regularization technique for solving such problems without pre-training. We re-formulate Kernel-PCA as an energy minimization problem in which low dimensionality constraints are introduced as regularization terms in the energy. To the best of our knowledge, ours is the first attempt to create a dimensionality regularizer in the KPCA framework. Our approach relies on robustly penalizing the rank of the recovered factors directly in the implicit feature space to create their low-dimensional approximations in closed form. Our approach performs robust KPCA in the presence of missing data and noise. We demonstrate state-of-the-art results on predicting missing entries in the standard oil flow dataset. Additionally, we evaluate our method on the challenging problem of Non-Rigid Structure from Motion and our approach delivers promising results on CMU mocap dataset despite the presence of significant occlusions and noise.
This paper considers an alternate formulation of Kernel PCA with rank constraints incorporated as a regularization term in the objective. The writing is not clear. The focus keeps shifting from estimating “causal factors”, to nonlinear dimensionality reduction to Kernel PCA to ill-posed inverse problems. The problem reformulation of Kernel PCA uses somewhat standard tricks and it is not clear what are the advantages of the proposed approach over the existing methods as there is no theoretical analysis of the overall approach or empirical comparison with existing state-of-the-art. - Not sure what the authors mean by “causal factors”. There is a reference to it in Abstract and in Problem formulation on page 3 without any definition/discussion. - In KPCA, I am not sure why one is interested in step (iii) outlined on page 2 of finding a pre-image for each - Authors outline two key disadvantages of the existing KPCA approach. The first one, that of low-dimensional manifold assumption not holding exactly, has received lots of attention in the machine learning literature. It is common to assume that the data lies near a low-dimensional manifold rather than on a low-dimensional manifold. Second disadvantage is somewhat unclear as finding “a data point (pre-image) corresponding to each projection in the input space” is not a standard step in KPCA. - On page 3, you never define , , . Clearly, they cannot be cartesian products. I have to assume that notation somehow implies N-tuples. - On page 3, Section 2, and are sets. What do you mean by - On page 5, is never defined. - Experiments: None of the standard algorithms for matrix completion such as OptSpace or SVT were considered - Experiments: There is no comparison with alternate existing approaches for Non-rigid structure from motion. - Proof of the main result Theorem 3.1: To get from (16) to (17) using the Holder inequality (as stated) one would end up with a term that involves sum of fourth powers of weights w_{ij}. Why would they equal to one using the orthonormal constraints? It would be useful to give more details here, as I don’t see how the argument goes through at this point.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Non-linear Dimensionality Regularizer for Solving Inverse Problems
Consider an ill-posed inverse problem of estimating causal factors from observations, one of which is known to lie near some (unknown) low-dimensional, non-linear manifold expressed by a predefined Mercer-kernel. Solving this problem requires simultaneous estimation of these factors and learning the low-dimensional representation for them. In this work, we introduce a novel non-linear dimensionality regularization technique for solving such problems without pre-training. We re-formulate Kernel-PCA as an energy minimization problem in which low dimensionality constraints are introduced as regularization terms in the energy. To the best of our knowledge, ours is the first attempt to create a dimensionality regularizer in the KPCA framework. Our approach relies on robustly penalizing the rank of the recovered factors directly in the implicit feature space to create their low-dimensional approximations in closed form. Our approach performs robust KPCA in the presence of missing data and noise. We demonstrate state-of-the-art results on predicting missing entries in the standard oil flow dataset. Additionally, we evaluate our method on the challenging problem of Non-Rigid Structure from Motion and our approach delivers promising results on CMU mocap dataset despite the presence of significant occlusions and noise.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



This paper considers an alternate formulation of Kernel PCA with rank constraints incorporated as a regularization term in the objective. The writing is not clear. The focus keeps shifting from estimating “causal factors”, to nonlinear dimensionality reduction to Kernel PCA to ill-posed inverse problems. The problem reformulation of Kernel PCA uses somewhat standard tricks and it is not clear what are the advantages of the proposed approach over the existing methods as there is no theoretical analysis of the overall approach or empirical comparison with existing state-of-the-art. - Not sure what the authors mean by “causal factors”. There is a reference to it in Abstract and in Problem formulation on page 3 without any definition/discussion. - In KPCA, I am not sure why one is interested in step (iii) outlined on page 2 of finding a pre-image for each - Authors outline two key disadvantages of the existing KPCA approach. The first one, that of low-dimensional manifold assumption not holding exactly, has received lots of attention in the machine learning literature. It is common to assume that the data lies near a low-dimensional manifold rather than on a low-dimensional manifold. Second disadvantage is somewhat unclear as finding “a data point (pre-image) corresponding to each projection in the input space” is not a standard step in KPCA. - On page 3, you never define , , . Clearly, they cannot be cartesian products. I have to assume that notation somehow implies N-tuples. - On page 3, Section 2, and are sets. What do you mean by - On page 5, is never defined. - Experiments: None of the standard algorithms for matrix completion such as OptSpace or SVT were considered - Experiments: There is no comparison with alternate existing approaches for Non-rigid structure from motion. - Proof of the main result Theorem 3.1: To get from (16) to (17) using the Holder inequality (as stated) one would end up with a term that involves sum of fourth powers of weights w_{ij}. Why would they equal to one using the orthonormal constraints? It would be useful to give more details here, as I don’t see how the argument goes through at this point.
assistant
This paper considers an alternate formulation of Kernel PCA with rank constraints incorporated as a regularization term in the objective. The writing is not clear. The focus keeps shifting from estimating “causal factors”, to nonlinear dimensionality reduction to Kernel PCA to ill-posed inverse problems. The problem reformulation of Kernel PCA uses somewhat standard tricks and it is not clear what are the advantages of the proposed approach over the existing methods as there is no theoretical analysis of the overall approach or empirical comparison with existing state-of-the-art. - Not sure what the authors mean by “causal factors”. There is a reference to it in Abstract and in Problem formulation on page 3 without any definition/discussion. - In KPCA, I am not sure why one is interested in step (iii) outlined on page 2 of finding a pre-image for each - Authors outline two key disadvantages of the existing KPCA approach. The first one, that of low-dimensional manifold assumption not holding exactly, has received lots of attention in the machine learning literature. It is common to assume that the data lies near a low-dimensional manifold rather than on a low-dimensional manifold. Second disadvantage is somewhat unclear as finding “a data point (pre-image) corresponding to each projection in the input space” is not a standard step in KPCA. - On page 3, you never define , , . Clearly, they cannot be cartesian products. I have to assume that notation somehow implies N-tuples. - On page 3, Section 2, and are sets. What do you mean by - On page 5, is never defined. - Experiments: None of the standard algorithms for matrix completion such as OptSpace or SVT were considered - Experiments: There is no comparison with alternate existing approaches for Non-rigid structure from motion. - Proof of the main result Theorem 3.1: To get from (16) to (17) using the Holder inequality (as stated) one would end up with a term that involves sum of fourth powers of weights w_{ij}. Why would they equal to one using the orthonormal constraints? It would be useful to give more details here, as I don’t see how the argument goes through at this point.