Using a Markov chain perspective of spectral clustering we present an algorithm to automatically find the number of stable clusters in a dataset. The Markov chain's behaviour is characterized by the spectral properties of the matrix of transition probabilities, from which we derive eigenflows along with their halflives. An eigenflow describes the flow of probabil- ity mass due to the Markov chain, and it is characterized by its eigen- value, or equivalently, by the halflife of its decay as the Markov chain is iterated. A ideal stable cluster is one with zero eigenflow and infi- nite half-life. The key insight in this paper is that bottlenecks between weakly coupled clusters can be identified by computing the sensitivity of the eigenflow's halflife to variations in the edge weights.

The manifold Helmholtzian (1-Laplacian) operator $\Delta_1$ elegantly generalizes the Laplace-Beltrami operator to vector fields on a manifold $\mathcal M$. In this work, we propose the estimation of the manifold Helmholtzian from point cloud data by a weighted 1-Laplacian $\mathbf{\mathcal L}_1$. While higher order Laplacians ave been introduced and studied, this work is the first to present a graph Helmholtzian constructed from a simplicial complex as an estimator for the continuous operator in a non-parametric setting. Equipped with the geometric and topological information about $\mathcal M$, the Helmholtzian is a useful tool for the analysis of flows and vector fields on $\mathcal M$ via the Helmholtz-Hodge theorem. In addition, the $\mathbf{\mathcal L}_1$ allows the smoothing, prediction, and feature extraction of the flows. We demonstrate these possibilities on substantial sets of synthetic and real point cloud datasets with non-trivial topological structures; and provide theoretical results on the limit of $\mathbf{\mathcal L}_1$ to $\Delta_1$.

In [1] we introduced a linear statistical model of joint color changes in images due to variation in lighting and certain non-geometric camera parameters. We did this by measuring the mappings of colors in one image of a scene to colors in another image of the same scene under different lighting conditions. Here we increase the flexibility of this color flow model by allowing flow coefficients to vary according to a low order polynomial over the image. This allows us to better fit smoothly varying lighting conditions as well as curved surfaces without endowing our model with too much capacity. We show results on image matching and shadow removal and detection.

Using a Markov chain perspective of spectral clustering we present an algorithm to automatically find the number of stable clusters in a dataset. The Markov chain's behaviour is characterized by the spectral properties of the matrix of transition probabilities, from which we derive eigenflows along with their halflives. An eigenflow describes the flow of probability mass due to the Markov chain, and it is characterized by its eigenvalue, or equivalently, by the halflife of its decay as the Markov chain is iterated. A ideal stable cluster is one with zero eigenflow and infinite half-life. The key insight in this paper is that bottlenecks between weakly coupled clusters can be identified by computing the sensitivity of the eigenflow's halflife to variations in the edge weights.

In [1] we introduced a linear statistical model of joint color changes in images due to variation in lighting and certain non-geometric camera parameters. We did this by measuring the mappings of colors in one image of a scene to colors in another image of the same scene under different lighting conditions. Here we increase the flexibility of this color flow model by allowing flow coefficients to vary according to a low order polynomial over the image. This allows us to better fit smoothly varying lighting conditions as well as curved surfaces without endowing our model with too much capacity. We show results on image matching and shadow removal and detection.

Using a Markov chain perspective of spectral clustering we present an algorithm to automatically find the number of stable clusters in a dataset. The Markov chain's behaviour is characterized by the spectral properties of the matrix of transition probabilities, from which we derive eigenflows along with their halflives. An eigenflow describes the flow of probability mass due to the Markov chain, and it is characterized by its eigenvalue, or equivalently, by the halflife of its decay as the Markov chain is iterated. A ideal stable cluster is one with zero eigenflow and infinite half-life. The key insight in this paper is that bottlenecks between weakly coupled clusters can be identified by computing the sensitivity of the eigenflow's halflife to variations in the edge weights.

In [1] we introduced a linear statistical model of joint color changes in images due to variation in lighting and certain non-geometric camera parameters.

Using a Markov chain perspective of spectral clustering we present an algorithm to automatically find the number of stable clusters in a dataset. The Markov chain's behaviour is characterized by the spectral properties of the matrix of transition probabilities, from which we derive eigenflows along with their halflives. An eigenflow describes the flow of probability massdue to the Markov chain, and it is characterized by its eigenvalue, orequivalently, by the halflife of its decay as the Markov chain is iterated. A ideal stable cluster is one with zero eigenflow and infinite half-life.The key insight in this paper is that bottlenecks between weakly coupled clusters can be identified by computing the sensitivity of the eigenflow's halflife to variations in the edge weights. We propose a novel EIGENCUTS algorithm to perform clustering that removes these identified bottlenecks in an iterative fashion.