[{"publication":"2016 Fourth International Conference on 3D Vision (3DV)","day":"19","_id":"18374","publication_identifier":{"isbn":["9781509054084"]},"doi":"10.1109/3dv.2016.53","oa":1,"type":"conference","citation":{"ieee":"A. M. Bronstein, Y. Choukroun, R. Kimmel, and M. Sela, “Consistent discretization and minimization of the L1 norm on manifolds,” in <i>2016 Fourth International Conference on 3D Vision (3DV)</i>,  Stanford, CA, United States, 2016.","chicago":"Bronstein, Alex M., Yoni Choukroun, Ron Kimmel, and Matan Sela. “Consistent Discretization and Minimization of the L1 Norm on Manifolds.” In <i>2016 Fourth International Conference on 3D Vision (3DV)</i>. IEEE, 2016. <a href=\"https://doi.org/10.1109/3dv.2016.53\">https://doi.org/10.1109/3dv.2016.53</a>.","short":"A.M. Bronstein, Y. Choukroun, R. Kimmel, M. Sela, in:, 2016 Fourth International Conference on 3D Vision (3DV), IEEE, 2016.","ama":"Bronstein AM, Choukroun Y, Kimmel R, Sela M. Consistent discretization and minimization of the L1 norm on manifolds. In: <i>2016 Fourth International Conference on 3D Vision (3DV)</i>. IEEE; 2016. doi:<a href=\"https://doi.org/10.1109/3dv.2016.53\">10.1109/3dv.2016.53</a>","mla":"Bronstein, Alex M., et al. “Consistent Discretization and Minimization of the L1 Norm on Manifolds.” <i>2016 Fourth International Conference on 3D Vision (3DV)</i>, 7785118, IEEE, 2016, doi:<a href=\"https://doi.org/10.1109/3dv.2016.53\">10.1109/3dv.2016.53</a>.","ista":"Bronstein AM, Choukroun Y, Kimmel R, Sela M. 2016. Consistent discretization and minimization of the L1 norm on manifolds. 2016 Fourth International Conference on 3D Vision (3DV). 4th International Conference on 3D Vision, 7785118.","apa":"Bronstein, A. M., Choukroun, Y., Kimmel, R., &#38; Sela, M. (2016). Consistent discretization and minimization of the L1 norm on manifolds. In <i>2016 Fourth International Conference on 3D Vision (3DV)</i>.  Stanford, CA, United States: IEEE. <a href=\"https://doi.org/10.1109/3dv.2016.53\">https://doi.org/10.1109/3dv.2016.53</a>"},"publication_status":"published","article_processing_charge":"No","oa_version":"Preprint","abstract":[{"lang":"eng","text":"The L 1 norm has been tremendously popular in signal and image processing in the past two decades due to its sparsity-promoting properties. More recently, its generalization to non-Euclidean domains has been found useful in shape analysis applications. For example, in conjunction with the minimization of the Dirichlet energy, it was shown to produce a compactly supported quasi-harmonic orthonormal basis, dubbed as compressed manifold modes [14]. The continuous L 1 norm on the manifold is often replaced by the vector ℓ 1 norm applied to sampled functions. We show that such an approach is incorrect in the sense that it does not consistently discretize the continuous norm and warn against its sensitivity to the specific sampling. We propose two alternative discretizations resulting in an iteratively-reweighed ℓ 2 norm. We demonstrate the proposed strategy on the compressed modes problem, which reduces to a sequence of simple eigendecomposition problems not requiring non-convex optimization on Stiefel manifolds and producing more stable and accurate results."}],"scopus_import":"1","year":"2016","extern":"1","month":"12","date_updated":"2024-12-05T14:19:34Z","arxiv":1,"publisher":"IEEE","date_published":"2016-12-19T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1609.05434","open_access":"1"}],"title":"Consistent discretization and minimization of the L1 norm on manifolds","quality_controlled":"1","status":"public","external_id":{"arxiv":["1609.05434"]},"author":[{"id":"58f3726e-7cba-11ef-ad8b-e6e8cb3904e6","last_name":"Bronstein","orcid":"0000-0001-9699-8730","first_name":"Alexander","full_name":"Bronstein, Alexander"},{"first_name":"Yoni","full_name":"Choukroun, Yoni","last_name":"Choukroun"},{"last_name":"Kimmel","full_name":"Kimmel, Ron","first_name":"Ron"},{"last_name":"Sela","full_name":"Sela, Matan","first_name":"Matan"}],"language":[{"iso":"eng"}],"date_created":"2024-10-15T11:20:54Z","conference":{"start_date":"2016-10-25","end_date":"2016-10-28","name":"4th International Conference on 3D Vision","location":" Stanford, CA, United States"},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","article_number":"7785118"},{"date_created":"2024-10-15T11:20:54Z","conference":{"name":"4th International Conference on 3D Vision","end_date":"2016-10-28","start_date":"2016-10-25","location":"Stanford, CA, United States"},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","article_number":"7785125","title":"SpectroMeter: Amortized sublinear spectral approximation of distance on graphs","quality_controlled":"1","status":"public","external_id":{"arxiv":["1609.05715"]},"author":[{"first_name":"Roee","full_name":"Litman, Roee","last_name":"Litman"},{"orcid":"0000-0001-9699-8730","last_name":"Bronstein","full_name":"Bronstein, Alexander","first_name":"Alexander","id":"58f3726e-7cba-11ef-ad8b-e6e8cb3904e6"}],"language":[{"iso":"eng"}],"publication_status":"published","article_processing_charge":"No","oa_version":"Preprint","extern":"1","scopus_import":"1","year":"2016","abstract":[{"text":"We present a method to approximate pairwise distance on a graph, having an amortized sub-linear complexity in its size. The proposed method follows the so called heat method due to Crane et al. The only additional input are the values of the eigenfunctions of the graph Laplacian at a subset of the vertices. Using these values we estimate a random walk from the source points, and normalize the result into a unit gradient function. The eigenfunctions are then used to synthesize distance values abiding by these constraints at desired locations. We show that this method works in practice on different types of inputs ranging from triangular meshes to general graphs. We also demonstrate that the resulting approximate distance is accurate enough to be used as the input to a recent method for intrinsic shape correspondence computation.","lang":"eng"}],"month":"12","arxiv":1,"date_updated":"2024-12-05T14:17:16Z","publisher":"IEEE","date_published":"2016-12-19T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1609.05715"}],"publication":"2016 Fourth International Conference on 3D Vision (3DV)","day":"19","_id":"18375","publication_identifier":{"isbn":["9781509054084"]},"doi":"10.1109/3dv.2016.60","oa":1,"citation":{"ama":"Litman R, Bronstein AM. SpectroMeter: Amortized sublinear spectral approximation of distance on graphs. In: <i>2016 Fourth International Conference on 3D Vision (3DV)</i>. IEEE; 2016. doi:<a href=\"https://doi.org/10.1109/3dv.2016.60\">10.1109/3dv.2016.60</a>","mla":"Litman, Roee, and Alex M. Bronstein. “SpectroMeter: Amortized Sublinear Spectral Approximation of Distance on Graphs.” <i>2016 Fourth International Conference on 3D Vision (3DV)</i>, 7785125, IEEE, 2016, doi:<a href=\"https://doi.org/10.1109/3dv.2016.60\">10.1109/3dv.2016.60</a>.","short":"R. Litman, A.M. Bronstein, in:, 2016 Fourth International Conference on 3D Vision (3DV), IEEE, 2016.","ista":"Litman R, Bronstein AM. 2016. SpectroMeter: Amortized sublinear spectral approximation of distance on graphs. 2016 Fourth International Conference on 3D Vision (3DV). 4th International Conference on 3D Vision, 7785125.","apa":"Litman, R., &#38; Bronstein, A. M. (2016). SpectroMeter: Amortized sublinear spectral approximation of distance on graphs. In <i>2016 Fourth International Conference on 3D Vision (3DV)</i>. Stanford, CA, United States: IEEE. <a href=\"https://doi.org/10.1109/3dv.2016.60\">https://doi.org/10.1109/3dv.2016.60</a>","ieee":"R. Litman and A. M. Bronstein, “SpectroMeter: Amortized sublinear spectral approximation of distance on graphs,” in <i>2016 Fourth International Conference on 3D Vision (3DV)</i>, Stanford, CA, United States, 2016.","chicago":"Litman, Roee, and Alex M. Bronstein. “SpectroMeter: Amortized Sublinear Spectral Approximation of Distance on Graphs.” In <i>2016 Fourth International Conference on 3D Vision (3DV)</i>. IEEE, 2016. <a href=\"https://doi.org/10.1109/3dv.2016.60\">https://doi.org/10.1109/3dv.2016.60</a>."},"type":"conference"}]