Exponential Space Research Articles

Multimodal and Multifactor Branching Time Active Inference.

Active inference is a state-of-the-art framework for modeling the brain that explains a wide range of mechanisms. Recently, two versions of branching time active inference (BTAI) have been developed to handle the exponential (space and time) complexity class that occurs when computing the prior over all possible policies up to the time horizon. However, those two versions of BTAI still suffer from an exponential complexity class with regard to the number of observed and latent variables being modeled. We resolve this limitation by allowing each observation to have its own likelihood mapping and each latent variable to have its own transition mapping. The implicit mean field approximation was tested in terms of its efficiency and computational cost using a dSprites environment in which the metadata of the dSprites data set was used as input to the model. In this setting, earlier implementations of branching time active inference (namely, BTAIVMP and BTAIBF) underperformed in relation to the mean field approximation (BTAI3MF) in terms of performance and computational efficiency. Specifically, BTAIVMP was able to solve 96.9% of the task in 5.1 seconds, and BTAIBF was able to solve 98.6% of the task in 17.5 seconds. Our new approach outperformed both of its predecessors by solving the task completely (100%) in only 2.559 seconds.

Approximate and Exact Optimization Algorithms for the Beltway and Turnpike Problems with Duplicated, Missing, Partially Labeled, and Uncertain Measurements.

The Turnpike problem aims to reconstruct a set of one-dimensional points from their unordered pairwise distances. Turnpike arises in biological applications such as molecular structure determination, genomic sequencing, tandem mass spectrometry, and molecular error-correcting codes. Under noisy observation of the distances, the Turnpike problem is NP-hard and can take exponential time and space to solve when using traditional algorithms. To address this, we reframe the noisy Turnpike problem through the lens of optimization, seeking to simultaneously find the unknown point set and a permutation that maximizes similarity to the input distances. Our core contribution is a suite of algorithms that robustly solve this new objective. This includes a bilevel optimization framework that can efficiently solve Turnpike instances with up to 100,000 points. We show that this framework can be extended to scenarios with domain-specific constraints that include duplicated, missing, and partially labeled distances. Using these, we also extend our algorithms to work for points distributed on a circle (the Beltway problem). For small-scale applications that require global optimality, we formulate an integer linear program (ILP) that (i) accepts an objective from a generic family of convex functions and (ii) uses an extended formulation to reduce the number of binary variables. On synthetic and real partial digest data, our bilevel algorithms achieved state-of-the-art scalability across challenging scenarios with performance that matches or exceeds competing baselines. On small-scale instances, our ILP efficiently recovered ground-truth assignments and produced reconstructions that match or exceed our alternating algorithms. Our implementations are available at https://github.com/Kingsford-Group/turnpikesolvermm.

Max-Product Sampling Kantorovich Operators: Quantitative Estimates in Functional Spaces

In this paper, we study the order of approximation for max-product sampling Kantorovich operators based upon generalized kernels in the setting of Orlicz spaces. We establish a quantitative estimate for the considered family of sampling-type operators using the Orlicz-type modulus of smoothness, which involves the modular functional of the space. From this result, it is possible to obtain the qualitative order of convergence when functions belonging to suitable Lipschitz classes are considered. On the other hand, in the compact case, we exploit a suitable definition of K-functional in Orlicz spaces in order to provide an upper bound for the approximation error of the involved operators. The treatment in the general framework of Orlicz spaces allows one to obtain a unifying theory on the rate of convergence, as the proved results can be deduced for a wide range of functional spaces, such as L p -spaces, interpolation spaces and exponential spaces.

Hybrid Modeling Techniques for Municipal Solid Waste Forecasting: An Application to OECD Countries

Abstract Accurate forecasting of municipal solid waste (MSW) generation is critical for effective waste management, given the rising volumes of waste posing environmental and public health challenges. This study investigates the efficacy of hybrid forecasting models in predicting MSW generation trends across Organization for Economic Cooperation and Development (OECD) countries. The empirical analysis utilizes five distinct approaches – ARIMA, Theta model, neural networks, exponential smoothing state space (ETS), and TBATS models. MSW data spanning 1995–2021 for 29 OECD nations are analyzed using the hybrid models and benchmarked against individual ARIMA models. The results demonstrate superior predictive accuracy for the hybrid models across multiple error metrics, capturing complex data patterns and relationships missed by individual models. The forecasts project continued MSW generation growth in most countries but reveal nuanced country-level differences as well. The implications for waste management policies include implementing waste reduction and recycling programs, investing in infrastructure and technology, enhancing public education, implementing pricing incentives, rigorous monitoring and evaluation of practices, and multi-stakeholder collaboration. However, uncertainties related to model selection and data limitations warrant acknowledgment. Overall, this study affirms the value of hybrid forecasting models in providing robust insights to inform evidence-based waste management strategies and transition toward sustainability in the OECD region.

Core allocation to minimize total flow time in a multicore system in the presence of a processing time constraint

Data centers are a vital and fundamental infrastructure component of the cloud. The requirement to execute a large number of demanding jobs places a premium on processing capacity. Parallelizing jobs to run on multiple cores reduces execution time. However, there is a decreasing marginal benefit to using more cores, with the speedup function quantifying the achievable gains. A critical performance metric is flow time. Previous results in the literature derived closed-form expressions for the optimal allocation of cores to minimize total flow time for a power-law speedup function if all jobs are present at time 0. However, this work did not place a constraint on the makespan. For many diverse applications, fast response times are essential, and latency targets are specified to avoid adverse impacts on user experience. This paper is the first to determine the optimal core allocations for a multicore system to minimize total flow time in the presence of a completion deadline (where all jobs have the same deadline). The allocation problem is formulated as a nonlinear optimization program that is solved using the Lagrange multiplier technique. Closed-form expressions are derived for the optimal core allocations, total flow time, and makespan, which can be fitted to a specified deadline by adjusting the value of a single Lagrange multiplier. Compared to the unconstrained problem, the shortest job first property for optimal allocation is maintained; however, a number of other properties require revising and other properties are only retained in a modified form (such as the scale-free and size-dependence properties). It is found that with a completion deadline the optimal solution may contain groups of simultaneous completions. In general, all possible patterns of single- and group-completion need to be considered, producing an exponential search space. However, the paper determines analytically that the optimal completion pattern consists of a sequence of single completions followed by a single group of simultaneous completions at the end, which reduces the search space dimension to being linear. The paper validates the Lagrange multiplier approach by verifying constraint qualifications.

Prescribed Performance Backstepping and Sliding Mode Control for Automated Platoons with IT2-Fuzzy Logic and Exponential Spacing Policy

Prescribed Performance Backstepping and Sliding Mode Control for Automated Platoons with IT2-Fuzzy Logic and Exponential Spacing Policy

Existence and multiplicity of solutions for a p(x)-Choquard-Kirchhoff problem involving critical growth and concave-convex nonlinearities

This paper is devoted to studying a kind of p(x)-Choquard-Kirchhoff problems involving critical growth and concave-convex nonlinearities. Combining the concentration-compactness principle for weighted variable exponent spaces, the calculus of variations, genus theory and the Hardy-Littlewood-Sobolev type inequality, we obtain the existence of nontrivial solutions for this kind of p(x)-Choquard-Kirchhoff problems. Our results improve the related ones in the literature.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Triple weak solution for p(x)-Laplacian like problem involving no flux boundary condition

Abstract In this paper, we consider a class of p ⁢ ( x ) {p(x)} -Laplacian like problems with indefinite weight involving no flux boundary condition. Using variational techniques and the critical point theorem of Bonanno and Marano [4], we prove the existence of at least three weak solutions to the problem in Sobolev variable exponent spaces.

Existence of solutions for singular elliptic equations with mixed boundary conditions

In this article, we investigate generalized solutions for elliptic equations involving double-phase ( p ( x ) , q ( x ) ) -Laplacian operators with Hardy potential in variable exponent spaces. ( p ( x ) , q ( x ) ) -Laplacian operators include p-Laplacian, q-Laplacian, p ( x ) -Laplacian and q ( x ) -Laplacian operators. Additionally, ( p ( x ) , q ( x ) ) -Laplacian elliptic equations with singular coefficients under mixed boundary conditions are seldom mentioned in previous work. Some new theorems of the existence on the generalized solutions are reestablished for such equations via variational methods when the nonlinearity satisfies suitable hypotheses in variable exponent Lebesgue spaces.

HTQ: Exploring the High-Dimensional Trade-Off of mixed-precision quantization

Mixed-precision quantization, where more sensitive layers are kept at higher precision, can achieve the trade-off between accuracy and complexity of neural networks. However, the search space for mixed-precision grows exponentially with the number of layers, making the brute force approach infeasible on deep networks. To reduce this exponential search space, recent efforts use Pareto frontier or integer linear programming to select the bit-precision of each layer. Unfortunately, we find that these prior works rely on a single constraint. In practice, model complexity includes space complexity and time complexity, and the two are weakly correlated, thus using simply one as a constraint leads to sub-optimal results. Besides this, they require manually set constraints, making them only pseudo-automatic. To address the above issues, we propose High-dimensional Trade-off Quantization (HTQ), which automatically determines the bit-precision in the high-dimensional space of model accuracy, space complexity, and time complexity without any manual intervention. Specifically, we use the saliency criterion based on connection sensitivity to indicate the accuracy perturbation after quantization, which performs similarly to Hessian information but can be calculated quickly (more than 1000× speedup). The bit-precision is then automatically selected according to the three-dimensional (3D) Pareto frontier of the total perturbation, model size, and bit operations (BOPs) without manual constraints. Moreover, HTQ allows for the joint optimization of weights and activations, and thus the bit-precisions of both can be computed concurrently. Compared to state-of-the-art methods, HTQ achieves higher accuracy and lower space/time complexity on various model architectures for image classification and object detection tasks. Code is available at: https://github.com/zkkli/HTQ.

Generalized Hölder Inequality in Herz-Morrey Spaces with Variable Exponent

The paper investigates the conditions for the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The main results are based on the exponent functions p(·) and α(·) . The proof of the first main result using the generalized Hölder’s inequality in Lebesgue spaces. The second main result of the paper is related to the weak space of the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The theorems state the equivalence of certain conditions for the inequality. Mathematical proofs and analysis are providing to support the presented results for findings contribute to the understanding of Hölder’s inequalities in variable exponent spaces and their applications in Herz-Morrey spaces.

Fixed Point of α-Modular Nonexpanive Mappings in Modular Vector Spaces 𝓁p(·)

Let C denote a convex subset within the vector space 𝓁p(·), and let T represent a mapping from C onto itself. Assume α=(α1,⋯,αn) is a multi-index in [0,1]n such that ∑i=1nαi=1, where α1>0 and αn>0. We define Tα:C→C as Tα=∑i=1nαiTi, known as the mean average of the mapping T. While every fixed point of T remains fixed for Tα, the reverse is not always true. This paper examines necessary and sufficient conditions for the existence of fixed points for T, relating them to the existence of fixed points for Tα and the behavior of T-orbits of points in T’s domain. The primary approach involves a detailed analysis of recurrent sequences in R. Our focus then shifts to variable exponent modular vector spaces 𝓁p(·), where we explore the essential conditions that guarantee the existence of fixed points for these mappings. This investigation marks the first instance of such results in this framework.

Mixed (regular and chaotic) dynamics under the channeling of the high energy positrons in [100] direction of the silicon crystal and Podolskiy-Narimanov distribution

The character of motion (regular or chaotic) on the quantum level manifests itself in the statistics of the set of the system's energy levels. The completely regular case generates the sequence of the levels with exponential (Poisson) level spacing distribution; the completely chaotic one — with Wigner distribution. The most interesting case is the co-existence of the regular and chaotic motion domains in the phase space of the system under consideration. The assumption of independent generation of two level sequencies by one chaotic domain and by all regular domains leads to Berry-Robnik distribution. However, the presence of the chaotic motion domain in the phase space affects the levels produced by the regular domains via the so-called chaos-assisted tunneling (CAT) that leads to Podolskiy-Narimanov distribution of the level spacings. This distribution needs the mean amplitude of the tunnel transition and the relative contribution of the regular domains to the mean level density as the parameters. Using their estimations for the transverse motion of the high energy positrons channeling near [100] direction in the silicon crystal we found that Podolskiy-Narimanov distribution demonstrates the best agreement with the level spacing distribution.

CuFastTucker: A Novel Sparse FastTucker Decomposition For HHLST on Multi-GPUs

High-order, high-dimension, and large-scale sparse tensors (HHLST) have found their origin in various real industrial applications, such as social networks, recommender systems, bioinformatics, and traffic information. To handle these complex tensors, sparse tensor decomposition techniques are employed to project the HHLST into a low-rank space. In this paper, we propose a novel sparse tensor decomposition model called Sparse FastTucker Decomposition (SFTD), which is a variant of Sparse Tucker Decomposition (STD). The SFTD utilizes Kruskal approximation for the core tensor, and we present a theorem that reduces the exponential space and computational overhead to a polynomial one. Additionally, we reduce the space overhead of intermediate parameters in the algorithmic process by sampling the intermediate matrix. Furthermore, this method guarantees convergence. To enhance the speed of SFTD, we leverage the compactness of matrix multiplication and parallel access through a stochastic strategy, resulting in GPU-accelerated cuFastTucker. Moreover, we propose a data division and communication strategy for cuFastTucker to accommodate data on Multi-GPU setups. Our proposed cuFastTucker demonstrates faster calculation and convergence speeds, as well as significantly lower space and computational overhead compared to state-of-the-art (SOTA) algorithms such as P-Tucker, Vest, GTA, Bigtensor and SGD_Tucker.

Convergence results in Orlicz spaces for sequences of max-product sampling Kantorovich operators

In this paper, we provide a unifying theory concerning the convergence properties of the so-called max-product sampling Kantorovich operators based upon generalized kernels in the setting of Orlicz spaces. The approximation of functions defined on both bounded intervals and on the whole real axis has been considered. Here, under suitable assumptions on the kernels, considered in order to define the operators, we are able to establish a modular convergence theorem for these sampling-type operators. As a direct consequence of the main theorem of this paper, we obtain that the involved operators can be successfully used for approximation processes in a wide variety of functional spaces, including the well-known interpolation and exponential spaces. This makes the Kantorovich variant of max-product sampling operators suitable for reconstructing not necessarily continuous functions (signals) belonging to a wide range of functional spaces. Finally, several examples of Orlicz spaces and of kernels for which the above theory can be applied, are presented.

Bilinear Calderón-Zygmund operator and its commutator on some variable exponent spaces of homogeneous type

Let $(X,d,\mu)$ be a space of homogeneous type in the sense of Coifman and and Weiss. In this setting, the author proves that a bilinear Calderon-Zygmund operator is bounded from the product of variable exponent Lebesgue spaces $L^{p_{1}(\cdot)}(X)\times L^{p_{2}(\cdot)}(X)$ into spaces $L^{p(\cdot)}(X)$, and it is bounded from the product of variable exponent generalized Morrey spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(X)\times \mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(X)$ into spaces $\mathcal{L}^{p(\cdot),\varphi}(X)$, where the Lebesgue measure functions $\varphi(\cdot,\cdot), \varphi_{1}(\cdot,\cdot)$ and $\varphi_{2}(\cdot,\cdot)$ satisfy $\varphi_{1}\times\varphi_{2}=\varphi$, and $\frac{1}{p(\cdot)}=\frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}$. Furthermore, by establishing sharp maximal estimate for the commutator $[b_{1},b_{2},BT]$ generated by $b_{1}, b_{2}\in\mathrm{BMO}(X)$ and $BT$, the author shows that the $[b_{1},b_{2},BT]$ is bounded from the product of spaces $L^{p_{1}(\cdot)}(X)\times L^{p_{2}(\cdot)}(X)$ into spaces $L^{p(\cdot)}(X)$, and it is also bounded from product of spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(X)\times \mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(X)$ into spaces $L^{p(\cdot),\varphi}(X)$.

Inclusion Relations Among Fractional Orlicz-Sobolev Spaces and a Littlewood-Paley Characterization

AbstractEmbeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. In particular, besides recovering standard embeddings for classical fractional Sobolev spaces, novel results are derived in borderline situations where the latter fail. For instance, limiting embeddings of Pohozhaev-Trudinger-Yudovich type into exponential spaces are offered. The equivalence of Gagliardo-Slobodeckij norms in fractional Orlicz-Sobolev spaces to norms defined via Littlewood-Paley decompositions, oscillations, or Besov type difference quotients is established as well. This equivalence, of independent interest, is a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.

Variable Herz–Morrey estimates for rough fractional Hausdorff operator

As a first attempt, we obtain the boundedness of the rough fractional Hausdorff operator on variable exponent Herz-type spaces. The method used in this paper enables us to study the operator on some other function spaces with variable exponents.

Connection Between Weighted Tail, Orlicz, Grand Lorentz And Grand Lebesgue Norms

We prove that the norm of functions in a suitable Grand Lorentz space built on a measure space, equipped with sigma finite diffuse measure, coincides with the norm in a suitable exponential Grand Lebesgue Space space as well as coincides with the so-called exponential tail norm, which may be quite described as norm in a suitable Banach rearrangement invariant space. We also exhibit comparisons with exponential Orlicz norms.