We orthogonalize the NSS model to condition and diagnose its ill-conditioned parameters.
problem The ill-conditioning of the NSS model's design matrix.
method Exact orthogonal reparametrization via QR decomposition.
result Orthogonalization isolates the conditioning structure and maintains fit uncertainty.
Proposes FOAGP for efficient orthogonal effect decomposition of black-box computer experiments.
problem Challenges in sensitivity analysis of black-box computer experiments with complex, nonlinear functional outputs.
method Functional-output orthogonal additive Gaussian process (FOAGP) with conditional orthogonality constraint.
result Demonstrates effectiveness in orthogonal effect decomposition and variance decomposition through simulations and real-world application.
We propose orthogonality as a necessary condition for disentangling aleatoric and epistemic uncertainty.
problem Jointly estimating aleatoric and epistemic uncertainty is problematic and non-trivial.
method We propose orthogonality as a necessary condition for disentanglement and construct UDE to measure orthogonality and consistency.
result Orthogonality and consistency are necessary and sufficient criteria for disentanglement.
Paper introduces GDR-learners for estimating potential outcomes from observational data.
problem Lack of theoretical property of general Neyman-orthogonality in deep generative models.
method Develops flexible GDR-learners based on various deep generative models.
result GDR-learners possess quasi-oracle efficiency and rate double robustness, asymptotically optimal.
Muon optimizer simplifies matrix optimization with spectral orthogonalization.
problem Matrix optimization challenges, especially with large condition numbers.
method Simplified Muon optimizer using spectral orthogonalization of gradients.
result Simplified Muon converges linearly with independent scalar sequences, outperforming gradient descent and Adam.
New convergence guarantees for learning with unknown nuisance parameters.
problem Learning problems with unknown nuisance parameters.
method Stochastic gradient optimization with Neyman orthogonality and approximately orthogonalized updates.
result Stochastic gradient algorithms can converge under conditions of nuisance parameters.
In this paper we study some properties of surfaces immersed in R4 whose asymptotic lines are orthogonal. We also analyze necessary and sufficient conditions for the hypersphericity of surfaces in R4.
OSA overcomes instability in skipless Transformers.
problem Instability in skipless Transformers using Softmax Self-Attention.
method OSA parametrizes attention matrix to be orthogonal via skew-symmetric matrix exponential.
result OSA allows for training non-causal Transformers without skip connections and normalisation layers.
PROD method improves high-dimensional regression by handling strong correlations.
problem Violation of Irrepresentable Condition in LASSO for high-dimensional data.
method PROD procedure based on orthogonal decomposition of design matrix.
result PROD enhances performance of high-dimensional penalized regression.
New algorithms improve tensor CP decomposition under mild conditions.
problem Improving tensor CP decomposition with theoretical guarantees under mild incoherence conditions.
method Composite PCA and Concurrent Orthogonalization algorithms.
result Theoretical guarantees and practical superiority over existing methods.
We consider the Orthogonal Least-Squares (OLS) algorithm for the recovery of a m-dimensional k-sparse signal from a low number of noisy linear measurements. The Exact Recovery Condition (ERC) in bounded noisy scenario is established for OLS under certain condition on nonzero elements of the signal. The new result a…
MuonEq improves training of matrix-valued parameters by rebalancing momentum before orthogonalization.
problem Training matrix-valued parameters with orthogonalized-update optimizers like Muon.
method MuonEq introduces three lightweight pre-orthogonalization equilibration schemes: two-sided row/column normalization (RC), row normalization (R), and column normalization (C).
result Row/column normalization acts as a zeroth-order surrogate for whitening and improves the geometry seen by orthogonalization.
The paper explores how semantic independence can be captured in text embeddings using partial orthogonality.
problem Capturing semantic independence in text embeddings.
method Developed a theory and methods based on partial orthogonality to demonstrate semantic independence.
result Partial orthogonality captures semantic independence in text embeddings.
We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provi…
In this short note we show the following result: Let (M2n+1,g) (n≥2) be a compact Sasaki manifold with positive transverse orthogonal bisectional curvature. Then π1(M) is finite, and the universal cover of (M2n+1,g) is isomorphic to a weighted Sasaki sphere. We also get some results in the case of n…
Orthogonal initialization does not speed up training in ultra-wide neural networks.
problem Exploring the effect of orthogonal initialization on training speed in deep neural networks.
method Study of neural tangent kernel dynamics in FCNs and CNNs with orthogonal initialization.
result The NTK of orthogonally-initialized networks remains constant during training, suggesting no speedup in the NTK regime.
Study spectral flow on a warped cylinder with special boundary conditions.
problem Analyzing spectral flow on a warped cylinder with specific boundary conditions.
method Complexifying the twisting bundle, diagonalizing the orthogonal twist, and regrouping conjugate and reflection-paired blocks.
result Explicit formula for RO(O(2))-valued spectral flow, refining ordinary spectral flow. The paper explores conditions for the existence of orthogonal almost complex structures on manifolds.
problem Conditions for the existence of orthogonal almost complex structures on manifolds.
method Analyzes the Nijenhuis tensor and its squared norm to determine the existence of orthogonal almost complex structures.
result There exists no orthogonal almost complex structure on the standard sphere \(S^6\) with \(|N|^2 < \frac{64}{5}\) everywhere.
AuON is a linear-time optimizer that improves upon Muon's performance without approximate orthogonal matrices.
problem High memory and computational costs of orthogonal momentum updates.
method AuON uses normalized nonlinear scaling and a 'emergency brake' to handle exploding attention logits.
result AuON achieves strong performance without approximate orthogonal matrices, preserving structural alignment and reconditioning.
In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are also obtained. The main new featu…
There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat …
Paper improves Monte Carlo sampling with new theoretical insights and methods.
problem Improving Monte Carlo sampling for variance reduction.
method Theoretical analysis of negatively dependent random variables and novel extensions using number theory and particle algorithms.
result Near-Orthogonal Monte Carlo (NOMC) consistently outperforms Orthogonal Monte Carlo (OMC) in various applications.
New algorithms learn sparse set functions in non-orthogonal Fourier bases.
problem Learning sparse set functions in non-orthogonal Fourier bases.
method Novel algorithms using non-orthogonal Fourier transforms.
result At most nk−klog2k+k queries for k non-zero Fourier coefficients. In this paper we study the class of compact Kähler manifolds with positive orthogonal Ricci curvature: Ric⊥>0. First we illustrate examples of Kähler manifolds with Ric⊥>0 on Kähler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of Kähler manifolds …
The paper solves PDEs from matrices with orthogonal columns, linking them to Hessian metrics and symmetric spaces.
problem Solving third order PDEs for strictly convex smooth functions.
method Geometric methods using Hessian metrics and symmetric spaces.
result Explicit solutions and a family of non-generic solutions with applications in Poisson geometry and Kahler structures.
New methods for estimating conditional odds and risk ratios improve treatment decision rules.
problem Estimation of conditional odds and risk ratios lags behind conditional average treatment effects.
method Proposed novel estimators based on doubly robust transformations and orthogonal risk functions.
result Proposed estimators significantly reduce bias and mean squared error in complex settings.
A new gradient boosting method improves interpretability of probabilistic models.
problem Learning interpretable yet accurate probabilistic models with limited rule complexity.
method A new objective function that measures the angle between risk gradient and condition output vector projection.
result Significantly improves comprehensibility/accuracy trade-off of fitted ensemble.
Novel method to quantify aleatoric uncertainty of treatment effects from observational data.
problem Understanding randomness in treatment effects for medical treatments.
method Partial identification and Neyman-orthogonality to quantify aleatoric uncertainty.
result Developed a novel orthogonal learner (AU-learner) for quantifying aleatoric uncertainty.
We obtain integral formulas for a metric-affine space equipped with two complementary orthogonal distributions. The integrand depends on the Ricci and mixed scalar curvatures and invariants of the second fundamental forms and integrability tensors of the distributions. The formulas under some conditions yield splitting…
New methods calibrate causal estimates using standard predictive models.
problem Calibrating causal treatment effect estimates.
method Developed algorithms to transform causal estimation into standard calibration.
result General algorithms for causal calibration using standard predictive models.
A well-conditioned Jacobian spectrum has a vital role in preventing exploding or vanishing gradients and speeding up learning of deep neural networks. Free probability theory helps us to understand and handle the Jacobian spectrum. We rigorously show almost sure asymptotic freeness of layer-wise Jacobians of deep neura…
Paper addresses group synchronization with incomplete measurements and proves linear convergence of GPM.
problem Orthogonal group synchronization with incomplete measurements and additive noise.
method Generalized power method (GPM) with local error bound analysis.
result Linear convergence of GPM to a global maximizer under general additive noise model.
Develops methods to estimate ratios of conditional expectation functions.
problem Estimating ratios of conditional expectation functions in causal inference.
method Orthogonal series estimator combined with debiased machine learning techniques.
result Valid pointwise and uniform asymptotic results for estimation and inference on CEFR.
Modified Laplacian connects to Yang-Mills instantons on manifolds.
problem Understanding instantons on 4D manifolds.
method Infinite dimensional Lévy Laplacian defined on manifolds, parameterized by curves in orthogonal rotations.
result Instantons on 4D manifolds are related to the modified Lévy Laplacian under specific curve conditions.
In this article, we study a class of Kähler manifolds defined on tube domains in Cn, and in particular those which have O(n)×Rn symmetry. For these, we prove a uniqueness result showing that any such manifold which is complete and has non-negative orthogonal bisectional curvature ($n \geq…
A classical problem in constant mean curvature hypersurface theory is, for given H≥0, to determine whether a compact submanifold Γn−1 of codimension two in Euclidean space R+n+1, having a single valued orthogonal projection on Rn, is the boundary of a graph with constant mean curvature H over a …
We propose the orthogonal random forest, an algorithm that combines Neyman-orthogonality to reduce sensitivity with respect to estimation error of nuisance parameters with generalized random forests (Athey et al., 2017)--a flexible non-parametric method for statistical estimation of conditional moment models using rand…
In this work, we find an equation that relates the Ricci curvature of a riemannian manifold M and the second fundamental forms of two orthogonal foliations of complementary dimensions, F and F⊥, defined on M. Using this equation, we show a sufficient condition for the manifold M to be …
A new knot invariant measures crossings in three orthogonal directions.
problem Defining a new knot invariant for certain knot diagrams.
method Defining the simultaneous crossing number for knots with doubly transvergent diagrams.
result The limit of the ratio of the new invariant to the usual crossing number is at most 8.
In this article we study compact K\ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive assuming local irreducibility. We also obtain a partial classification of possib…
New method solves sparse PCA for multiple components efficiently.
problem Sparse PCA for multiple orthogonal components.
method Reformulates orthogonality as rank constraints, uses semidefinite relaxations and bounds.
result Exact solutions with near-optimal variance explained and orthogonality.
This paper simplifies conditional Sobol' indices calculation using PCE bases.
problem Computational inefficiency and lack of consistency in evaluating conditional Sobol' indices.
method Analytical extraction of conditional Sobol' indices via basis decomposition of PCE expansions.
result Derives closed-form expressions for conditional Sobol' indices.
In this paper, we present new results on using orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries for complex cases (i.e., complex measurement vector, complex dictionary and complex additive white Gaussian noise (CAWGN)). A sufficient condition that OMP can recover …
Wasserstein-GANs have been introduced to address the deficiencies of generative adversarial networks (GANs) regarding the problems of vanishing gradients and mode collapse during the training, leading to improved convergence behaviour and improved image quality. However, Wasserstein-GANs require the discriminator to be…
In this paper, we will give an extension of Mok's theorem on the generalized Frankel conjecture under the condition of the orthogonal bisectional curvature.
New method for sampling on constrained domains using orthogonal-space gradient flow.
problem Sampling on manifolds defined by constraints is challenging.
method Orthogonal-Space Variational Gradient Descent (O-Gradient)
result O-Gradient converges to the target constrained distribution efficiently.
The paper extends orthogonal decomposition results to hermitian Higgs bundles.
problem Classical propositions on holomorphic vector bundles do not always extend to Higgs bundles.
method The approach involves extending propositions on orthogonal decompositions and the second fundamental form to hermitian Higgs bundles.
result Extended propositions concerning orthogonal decompositions and the second fundamental form have applications in Higgs bundles.
A new method tests Expected Shortfall by analyzing both duration and severity of VaR violations.
problem Lack of separate testing for frequency and severity in ES backtesting.
method Uses bivariate orthogonal polynomials to derive moment conditions for durations and severities.
result Proposes a Wald test for identifying mis-specified components in ES models.