Size matters: cardinality-constrained clustering and outlier detection via conic optimization

Plain vanilla K-means clustering has proven to be successful in practice, yet it suffers from outlier sensitivity and may produce highly unbalanced clusters. To mitigate both shortcomings, we formulate a joint outlier detection and clustering problem, which assigns a prescribed number of datapoints to an auxiliary outlier cluster and performs cardinality-constrainedK-means clustering on the residual dataset, treating the cluster cardinalities as a given input. We cast this problem as a mixed-integer linear program (MILP) that admits tractable semidefinite and linear programming relaxations. We propose deterministic rounding schemes thattransform the relaxed solutions to feasible solutions for the MILP. We also prove that these solutions areoptimal in the MILP if a cluster separation condition holds

In Perfect Shape: Certifiably Optimal 3D Shape Reconstruction from 2D Landmarks

We study the problem of 3D shape reconstruction from 2D landmarks extracted in a single image. We adopt the 3D deformable shape model and formulate the reconstruction as a joint optimization of the camera pose and the linear shape parameters. Our first contribution is to apply Lasserre's hierarchy of convex Sums-of-Squares (SOS) relaxations to solve the shape reconstruction problem and show that the SOS relaxation of minimum order 2 empirically solves the original non-convex problem exactly. Our second contribution is to exploit the structure of the polynomial in the objective function and find a reduced set of basis monomials for the SOS relaxation that significantly decreases the size of the resulting semidefinite program (SDP) without compromising its accuracy. These two contributions, to the best of our knowledge, lead to the first certifiably optimal solver for 3D shape reconstruction, that we name Shape*. Our third contribution is to add an outlier rejection layer to Shape* using a truncated least squares (TLS) robust cost function and leveraging graduated non-convexity to solve TLS without initialization. The result is a robust reconstruction algorithm, named Shape#, that tolerates a large amount of outlier measurements. We evaluate the performance of Shape* and Shape# in both simulated and real experiments, showing that Shape* outperforms local optimization and previous convex relaxation techniques, while Shape# achieves state-of-the-art performance and is robust against 70% outliers in the FG3DCar dataset.Comment: Camera-ready, CVPR 2020. 18 pages, 5 figures, 1 tabl

SIM-Sync: From Certifiably Optimal Synchronization over the 3D Similarity Group to Scene Reconstruction with Learned Depth

This paper presents SIM-Sync, a certifiably optimal algorithm that estimates camera trajectory and 3D scene structure directly from multiview image keypoints. SIM-Sync fills the gap between pose graph optimization and bundle adjustment; the former admits efficient global optimization but requires relative pose measurements and the latter directly consumes image keypoints but is difficult to optimize globally (due to camera projective geometry). The bridge to this gap is a pretrained depth prediction network. Given a graph with nodes representing monocular images taken at unknown camera poses and edges containing pairwise image keypoint correspondences, SIM-Sync first uses a pretrained depth prediction network to lift the 2D keypoints into 3D scaled point clouds, where the scaling of the per-image point cloud is unknown due to the scale ambiguity in monocular depth prediction. SIM-Sync then seeks to synchronize jointly the unknown camera poses and scaling factors (i.e., over the 3D similarity group). The SIM-Sync formulation, despite nonconvex, allows designing an efficient certifiably optimal solver that is almost identical to the SE-Sync algorithm. We demonstrate the tightness, robustness, and practical usefulness of SIM-Sync in both simulated and real experiments. In simulation, we show (i) SIM-Sync compares favorably with SE-Sync in scale-free synchronization, and (ii) SIM-Sync can be used together with robust estimators to tolerate a high amount of outliers. In real experiments, we show (a) SIM-Sync achieves similar performance as Ceres on bundle adjustment datasets, and (b) SIM-Sync performs on par with ORB-SLAM3 on the TUM dataset with zero-shot depth prediction.Comment: 28 page