26 problems

Flow matching with linear interpolation is trained using an interpolation-based objective, and its convergence rate has a support-estimation component that is slower than the optim…

Let the ambient input space be the space containing real-world datasets, and let the intrinsic dimensionality of a dataset mean the dimensionality of the manifold on which it lies.…

Let $M$ be a sufficiently smooth unknown manifold, and consider spectrum estimation for a weighted Laplace operator on $M$. Unknown-manifold rate conjecture. Lack of direct access…

Let $M$ be an unknown sufficiently smooth $d$-dimensional submanifold of $\mathbb{R}^D$. Aamari and Levrard's chart-based manifold estimators can be adapted to construct an estimat…

Let ${\mathsf{lPCA}}\xspace$ denote a local principal component analysis estimator that infers the dimension of the tangent plane, and let ${\mathsf{MLE}}\xspace$ denote a parametr…

Intrinsic-dimension conjecture. In general, the parameter $d$ can be replaced by $d_{\mathrm{int}}$ in all the results: tempered overfitting occurs whenever…

Let $G=(V,E,w)$ be a graph with features given by a map $f:V\to M$ into a Riemannian manifold, and suppose the graph's features have a convex bounding ball. Convergence conjecture.…

Manifold reconstruction conjecture. Given the Manifold Hypothesis that time series data lie on a manifold, this method reconstructs this topological space from the input time serie…

High-dimensional datasets encountered in practice are expected to concentrate near smooth manifolds of low intrinsic dimension. Manifold hypothesis. Such datasets tend to concentra…

The framework models a data set $\{x_1,\dots,x_N\}\subset\mathbb{R}^d$ using a tangent subbundle consisting of affine subspaces of $\mathbb{R}^d$, with a kernel range parameter con…

Let ${\mathbb S}^1_A \times {\mathbb S}^1_B$ be a flat product torus with $A>B$, and let $\psi^{\lambda\,\textrm{ near }(k/B)^2}$ denote an approximately selective embedding into…

Higher-dimensional convergence conjecture. This property should also hold in more general settings and for higher intrinsic dimensions not covered by the available theorem.

Let $\mathcal{M}$ be a smooth orientable $(D-1)$-dimensional manifold in $\mathbb{R}^D$. For each chart $U_i$, let $N_i:U_i\to\mathcal{S}^{D-1}$ be its Gauss map. Suppose there exi…

Let $\mathcal M$ be the manifold in the convergence analysis, and let $\varepsilon$ denote the truncation parameter. The convergence bound above has a dependence on $\varepsilon$ o…

Manifold hypothesis. The dataset lies on a low-dimensional submanifold with high probability.

Let $M$ be a lower-dimensional manifold, let $G_n$ be the graph used to define the graph distance $d_{G_n}$, and let $d_M$ denote the distance on $M$. Graph-distance underestimatio…

Deep-network extension conjecture. Theorem 1 should extend to very deep fully connected neural network classes in which the number of neurons per layer is fixed and the number of h…

A dynamical system is said to have compact solutions with a low number of unstable directions when its relevant solutions remain compact and the number of unstable directions is sm…

Let $f$ satisfy assumptions (A1)$_{f,6}$, (A2)$_f$, and (A3)$_f$, and let $R_a(f)$ be the set of limits of the Euler iterates … for $x\in\partial S_\epsilon$, where…

General certificate conjecture. A phenomenon analogous to the existence of a certificate for the two-circles geometry should hold for general one-dimensional manifolds, possibly wi…

Mean-map injectivity conjecture. For the stochastic dynamical system considered here, no two transition densities $p^\tau(x_1,\cdot)$ and $p^\tau(x_2,\cdot)$ possess the same mean;…

Graph-posterior convergence conjecture. The convergence of the graph posteriors to the ground truth posterior, together with guidelines for choosing the kernel bandwidth optimally…

Let $M$ be a manifold with boundary, let $M_w$ denote the wave region associated with the LLE asymptotic operator, and let $\mathcal{D}_\epsilon$ be the corresponding LLE operator…

Let $\lambda_i$ be the eigenvalues appearing in the limiting distribution … where the $\chi_{1,i}^2$ are independent chi-squared random variables with one degree of freedom. Gaussi…

Let $(\mathcal{M},g(\cdot))$ be the underlying manifold, and let $[0,T]$ be the time interval over which the time coupled diffusion map is constructed from the time-dependent data…