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Found 22 papers, 5 papers with code
Randomly Pivoted Partial Cholesky: Random How?
We consider the problem of finding good low rank approximations of symmetric, positive-definite $A \in \mathbb{R}^{n \times n}$.
May the force be with you
Modern methods in dimensionality reduction are dominated by nonlinear attraction-repulsion force-based methods (this includes t-SNE, UMAP, ForceAtlas2, LargeVis, and many more).
A common variable minimax theorem for graphs
Let $\mathcal{G} = \{G_1 = (V, E_1), \dots, G_m = (V, E_m)\}$ be a collection of $m$ graphs defined on a common set of vertices $V$ but with different edge sets $E_1, \dots, E_m$.
t-SNE, Forceful Colorings and Mean Field Limits
t-SNE is one of the most commonly used force-based nonlinear dimensionality reduction methods.
Max-Cut via Kuramoto-type Oscillators
Burer, Monteiro & Zhang proposed to find, for $n$ angles $\left\{\theta_1, \theta_2, \dots, \theta_n\right\} \subset [0, 2\pi]$, minima of the energy $$ f(\theta_1, \dots, \theta_n) = \sum_{i, j=1}^{n} a_{ij} \cos{(\theta_i - \theta_j)}$$ because configurations achieving a global minimum leads to a partition of size 0. 878 Max-Cut(G).
Optimization and Control Data Structures and Algorithms
Hessian Estimates for Laplacian Eigenfunctions
We study Laplacian eigenfunctions $-\Delta \phi_k = \lambda_k \phi_k$ with a Dirichlet condition on bounded domains $\Omega \subset \mathbb{R}^n$ with smooth boundary.
Analysis of PDEs
Neural Collapse with Cross-Entropy Loss
We consider the variational problem of cross-entropy loss with $n$ feature vectors on a unit hypersphere in $\mathbb{R}^d$.
Finding Structure in Sequences of Real Numbers via Graph Theory: a Problem List
We investigate a method of generating a graph $G=(V, E)$ out of an ordered list of $n$ distinct real numbers $a_1, \dots, a_n$.
Combinatorics
On the Regularization Effect of Stochastic Gradient Descent applied to Least Squares
We study the behavior of stochastic gradient descent applied to $\|Ax -b \|_2^2 \rightarrow \min$ for invertible $A \in \mathbb{R}^{n \times n}$.
Spectral Clustering Revisited: Information Hidden in the Fiedler Vector
We are interested in the clustering problem on graphs: it is known that if there are two underlying clusters, then the signs of the eigenvector corresponding to the second largest eigenvalue of the adjacency matrix can reliably reconstruct the two clusters.
The Spectral Underpinning of word2vec
word2vec due to Mikolov \textit{et al.} (2013) is a word embedding method that is widely used in natural language processing.
Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations
T-distributed stochastic neighbour embedding (t-SNE) is a widely used data visualisation technique.
Recovering Trees with Convex Clustering
Convex clustering refers, for given $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^p$, to the minimization of \begin{eqnarray*} u(\gamma) & = & \underset{u_1, \dots, u_n }{\arg\min}\;\sum_{i=1}^{n}{\lVert x_i - u_i \rVert^2} + \gamma \sum_{i, j=1}^{n}{w_{ij} \lVert u_i - u_j\rVert},\\ \end{eqnarray*} where $w_{ij} \geq 0$ is an affinity that quantifies the similarity between $x_i$ and $x_j$.
On the Dual Geometry of Laplacian Eigenfunctions
We discuss the geometry of Laplacian eigenfunctions $-\Delta \phi = \lambda \phi$ on compact manifolds $(M, g)$ and combinatorial graphs $G=(V, E)$.
Numerical Integration on Graphs: where to sample and how to weigh
Let $G=(V, E, w)$ be a finite, connected graph with weighted edges.
Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding
t-distributed Stochastic Neighborhood Embedding (t-SNE) is a method for dimensionality reduction and visualization that has become widely popular in recent years.
Randomized Near Neighbor Graphs, Giant Components, and Applications in Data Science
If we pick $n$ random points uniformly in $[0, 1]^d$ and connect each point to its $k-$nearest neighbors, then it is well known that there exists a giant connected component with high probability.
Clustering with t-SNE, provably
t-distributed Stochastic Neighborhood Embedding (t-SNE), a clustering and visualization method proposed by van der Maaten & Hinton in 2008, has rapidly become a standard tool in a number of natural sciences.
The Geometry of Nodal Sets and Outlier Detection
Let $(M, g)$ be a compact manifold and let $-\Delta \phi_k = \lambda_k \phi_k$ be the sequence of Laplacian eigenfunctions.
Stochastic Neighbor Embedding separates well-separated clusters
Stochastic Neighbor Embedding and its variants are widely used dimensionality reduction techniques -- despite their popularity, no theoretical results are known.
On the Diffusion Geometry of Graph Laplacians and Applications
We study directed, weighted graphs $G=(V, E)$ and consider the (not necessarily symmetric) averaging operator $$ (\mathcal{L}u)(i) = -\sum_{j \sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights.
Spectral Echolocation via the Wave Embedding
Spectral embedding uses eigenfunctions of the discrete Laplacian on a weighted graph to obtain coordinates for an embedding of an abstract data set into Euclidean space.
