The paper optimizes identifying top k arms from a fraction of ρ arms in stochastic bandits.
problem Identifying k distinct arms among the top ρ fraction of arms in stochastic bandits with a PAC tolerance. method The paper considers two cases: known and unknown threshold of top arms' expected rewards. It proves lower bounds and proposes algorithms for each case, showing sample complexity optimality for two algorithms.
result Two algorithms are sample complexity optimal (up to constant factors) and the other two are optimal up to a log factor.
Dropping a tiny fraction of preferences can significantly alter the rankings of top LLMs.
problem Robustness of LLM ranking systems to small changes in preference data.
method A computational method based on the Bradley-Terry model to evaluate robustness.
result Top LLM rankings can be highly sensitive to the removal of a small fraction of preferences.
In this paper we show that there are applications that transform the movement of a pendulum into movements in R3. This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top system reduces to the equation of a pendulum. Those properties are also considered in…
Socio-economic inequality is characterized from data using various indices. The Gini (g) index, giving the overall inequality is the most common one, while the recently introduced Kolkata (k) index gives a measure of 1−k fraction of population who possess top k fraction of wealth in the society. Here, we show t…
A method for finding most influential sets reduces a complex problem to a sequence of simpler top-k problems.
problem Identifying most influential subsets in complex models.
method Reduces the problem to a sequence of top-k problems using Dinkelbach's method. result The method returns a globally optimal set for the univariate ratio objective, including partial linear models.
Socio-economic inequality is measured using various indices. The Gini (g) index, giving the overall inequality is the most commonly used, while the recently introduced Kolkata (k) index gives a measure of 1−k fraction of population who possess top k fraction of wealth in the society. This article reviews the ch…
The purpose if this master's thesis is to study and develop a new algorithmic framework for Collaborative Filtering to produce recommendations in the top-N recommendation problem. Thus, we propose Lanczos Latent Factor Recommender (LLFR); a novel "big data friendly" collaborative filtering algorithm for top-N recommend…
Study on cohomology of special linear groups over Euclidean number rings.
problem Understanding cohomology of principal congruence subgroups of special linear groups.
method Analyzing cohomology groups and using properties of Tits buildings.
result Natural map between cohomology and homology is surjective and an isomorphism under certain conditions.
Through simple analytical calculations and numerical simulations, we demonstrate the generic existence of a self-organized macroscopic state in any large multivariate system possessing non-vanishing average correlations between a finite fraction of all pairs of elements. The coexistence of an eigenvalue spectrum predic…
A new method for efficiently updating large-scale matrices in real-time.
problem Updating large-scale matrices with evolving data in real-time.
method Incremental SVD approach that handles row/column appends, rank-1 updates, and refresh strategies.
result Incremental SVD achieves accuracy close to full SVD with a fraction of the computational cost.
Spectral clustering is a technique that clusters elements using the top few eigenvectors of their (possibly normalized) similarity matrix. The quality of spectral clustering is closely tied to the convergence properties of these principal eigenvectors. This rate of convergence has been shown to be identical for both th…
In this work we present a new approach on studying dynamical systems. Combining the two ways of expressing the uncertainty, using probabilistic theory and credibility theory, we have research the generalized fractional hybrid equations. We have introduced the concepts of generalized fractional Wiener process, generaliz…
Projective resolves symplectic Steinberg module for number rings.
problem Constructing a projective resolution for symplectic Steinberg module.
method Similar to special linear group, but more complex construction.
result Computed top degree cohomology of congruence subgroups.
Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and as…
Introduces fractional k-dimensional measure bridging fractional length and area.
problem Defining fractional measures for dimensions between 0 and n-1.
method Introduces a parameterized fractional measure σ that converges to Hausdorff measure. result Fractional measure converges to Hausdorff measure with a known constant factor.
The theory of derivative of noninteger order goes back to Leibniz, Liouville and Riemann. Derivatives of fractional order have found many applications in recent studies in mechanics, physics, economics. In this paper we define the fractional tangent bundle on a manifold, using a method of Radu Miron. The fractional Lei…
Let Sg be a closed orientable surface of genus g≥2 and C a simple closed nonseparating curve in F. Let tC denote a left handed Dehn twist about C. A \textit{fractional power} of tC of \textit{exponent} $\fraction{\ell}{n}$ is an $h \in \Mod(S_g)$ such that hn=tCℓ. Unlike a root of a $t…
Study on fractional curvature flow on unit sphere, extending previous work.
problem Fractional Nirenberg problem on unit sphere.
method Fractional conformal curvature flow on unit sphere.
result Perturbation result for fractional Nirenberg problem with σ∈(1/2,1). We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman's functionals and derived the corresponding fractional evolution …
Using Caputo fractional derivative of order α we build the fractional jet bundle of order α and its main geometrical structures. Defined on that bundle, some fractional dynamical systems with applications to economics are studied.
Paper analyzes trade-offs in top-k classification accuracies and proposes a new loss function.
problem CE loss does not always optimize top-k prediction, especially with complex data.
method Introduces a novel top-k transition loss to improve top-k accuracy.
result Our loss function improves top-k accuracy, especially for k > 10.
Volatility roughness studied using fractional noise-driven models.
problem Volatility roughness interpretation.
method Data-reconstructed fractional volatility model with fractional noise.
result Option pricing equation and solution derived using Malliavin calculus.
Using the reviewed Riemann-Liouville fractional derivative we introduce the fractional osculator Lagrange space of k order and the main structures on it. The results are applied at the k order fractional prolongation of Lagrange, Finsler and Riemann fractional structures.
We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this int…
Introduces fractional length and nonlocal curvature for smooth curves.
problem Defining curvature for curves of fractional length.
method Introduces fractional length and derives nonlocal curvature using fractional perimeter analogy.
result Fractional length converges to traditional length with a multiplicative constant.
Develops fractional de Rham theory for Maxwell equations.
problem Formulating fractional calculus for Maxwell equations.
method Fractional tangent functionals, Riemann-Liouville integral, polynomial algebra, exterior algebra.
result Fractional de Rham complex for Maxwell equations.
Modeling financial markets with memory using fractional calculus and Brownian motion.
problem Capturing memory effects in financial markets using stochastic models.
method Fractional Langevin equation with colored noise generated by fractional Brownian motion.
result Anomalous marginal glass phase observed in some regions of the system.
Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(…
We study the fractional gravity for spacetimes with non-integer dimensions. Our constructions are based on a geometric formalism with the fractional Caputo derivative and integral calculus adapted to nonolonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical …
Extends fractional Lp uncertainty principles with extremizers and stability results.
problem Investigating uncertainty principles in fractional Lp settings. method Analyzing the fractional Schrödinger equation to find extremal functions and sharp constants.
result Proves stability of extremizers for fractional uncertainty inequalities.
In this paper we established the condition for a curve to satisfy stochas- tic fractional HP (Hamilton-Pontryagin) equations. These equations are described using It^o integral. We have also considered the case of stochastic fractional Hamiltonian equa- tions, for a hyperregular Lagrange function. From the stochastic fr…
Approximates derivative pricing under fractional stochastic volatility.
problem Derivative pricing under fractional stochastic volatility model.
method Approximate expression derived from deterministic functions and fractional Ornstein-Uhlenbeck process.
result Numerical simulations show the feasibility and effect of long-range dependencies on derivative prices.
New method uses fractional posteriors for semiparametric inference with improved uncertainty quantification.
problem Semiparametric inference with nonparametric priors and fractional posteriors.
method Established a general Bernstein--von Mises theorem for fractional posterior distributions, proposed shifted-and-rescaled credible sets.
result Fractional posterior credible sets provide reliable uncertainty quantification but have inflated size; shifted-and-rescaled set is an efficient confidence set.
Derives an option-pricing formula for fractional markets with skew and smile.
problem Developing a pricing formula for financial options with skew and smile.
method Employed the Lévy-Khintchine theorem and fractional Gaussian noise to generalize the Black-Scholes-Merton formula.
result An exponentially convergent option-pricing formula for fractional markets.
In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of continued fractions with arbitrary elements.
The paper studies inequalities for fractional GJMS operators on conformal infinity.
problem Deriving comparison inequalities for fractional Yamabe constants.
method Using Poincaré-Einstein manifolds and fractional GJMS operators.
result Two comparison inequalities for fractional Yamabe constants are derived.
Using the fractional integration and differentiation on R we build the fractional jet fibre bundle on a differentiable manifold and we emphasize some important geometrical objects. Euler-Lagrange fractional equations are described. Some significant examples from mechanics and economics are presented.
The paper analyzes top-k classification and proposes consistent loss functions.
problem Understanding consistency of top-k classification in challenging tasks.
method Theoretical analysis, defining top-k calibration, proposing new loss functions.
result Proposes a new consistent hinge loss and a top-k calibrated convex loss.
Top/O's first two k-invariants are zero.
problem Understanding the structure of Top/O.
method Analyzing the first two k-invariants.
result The first two k-invariants of Top/O vanish.
New model uses generalized fractional Brownian motion for stock price prediction.
problem Traditional models fail to accurately predict stock price fluctuations.
method Introduces generalized fractional Brownian motion as a new stochastic process for price modeling.
result Validates the new model for option pricing and risk assessment.
New framework for ranking distributions using variable fractional parameters.
problem Ordering distributions with varying steepness and local non-concavities.
method Introducing a function γ:Ro[0,1] to replace the fixed parameter in fractional SD. result Enables ranking of a broader range of distributions and incorporates dynamic greediness.
Study proves smooth solutions for fractional mean curvature flow within short time.
problem Short-time existence of smooth solutions for fractional mean curvature flow.
method Established using short-time existence theorem for bounded, C^{1,1}-regular initial sets.
result Smooth solutions exist for both fractional mean curvature flow and volume preserving flow.
Paper approximates fractional harmonic maps with numerical methods.
problem Approximating fractional harmonic maps with constraints and nonlocality.
method Weak compactness results and numerical methods for various PDEs.
result Convergence of numerical approximations for fractional harmonic maps.
The mixed-fractional CEV model improves CDS pricing by accounting for default risk.
problem Improving the pricing of Credit Default Swaps (CDS) by accounting for default risk.
method Using a mixed-fractional Brownian motion to model the Constant Elasticity of Variance (CEV) model.
result The mixed-fractional CEV model yields more realistic CDS spreads and default probabilities.
Fractional combinatorial flow improves surface conformal structures.
problem Improving discrete conformal structures on surfaces.
method Introducing a fractional combinatorial Calabi flow for discrete conformal structures on surfaces.
result Longtime existence and global convergence of the fractional combinatorial Calabi flow for various surface types.
This note develops certain sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter.
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The def…
Fractional Sobolev metrics on immersions are well-posed.
problem Analyzing fractional Sobolev metrics on immersions.
method Proving geodesic equations are locally well-posed.
result Fractional Sobolev metrics on spaces of immersions have well-posed geodesic equations.