A method for constructing a polynomial markov operator based on ishikawa iteration
By constructing a polynomial Markov operator through Ishikawa iteration, the problems of small spectral gaps and slow convergence speed in the Markov chain Monte Carlo method are solved, and efficient high-dimensional Bayesian inference and complex distribution sampling are achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- GUILIN UNIV OF ELECTRONIC TECH
- Filing Date
- 2026-03-27
- Publication Date
- 2026-06-09
AI Technical Summary
Existing Markov chain Monte Carlo methods suffer from problems such as small spectral gaps in the transition operators, slow convergence speed, and high asymptotic variance.
By introducing the Ishikawa iterative polynomial Markov operator construction method, and employing a two-stage iterative structure and multi-step convex combination, a polynomial Markov operator with improved spectral properties is constructed. This includes the construction of the basic Markov operator, the introduction of the Ishikawa iterative structure, the construction of the polynomial Markov operator, and the analysis of the operator properties.
While keeping the target distribution unchanged, this method effectively improves the spectral properties of Markov chains, increases sampling efficiency, reduces asymptotic variance, and enhances convergence speed.
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Abstract
Description
Technical Field
[0002] This work relates to the fields of stochastic processes and Monte Carlo computation, and in particular to a method for constructing a polynomial Markov transition operator based on Ishikawa iteration. Background Technology
[0004] Markov chain Monte Carlo (MCMC) methods, by constructing Markov chains with the target distribution as an invariant measure, enable sampling of complex high-dimensional distributions and have become a core tool in Bayesian inference, statistical physics, and scientific computing. [1 ][2][3] These techniques are particularly well-suited for quantifying uncertainties in high-dimensional parameter spaces, such as sampling from the posterior distribution in Bayesian models to support decision-making and prediction. However, the classic Metropolis–Hastings (MH) random walk algorithm often suffers from slow mixing and high sample autocorrelation, leading to a reduced effective sample size (ESS), especially in multimodal or long-tailed objectives, resulting in increased Monte Carlo variance. [1][2][3] This limitation is particularly evident in the era of big data, where high dimensionality and computational complexity exacerbate autocorrelation effects, requiring more iterations to reach stability.
[0005] To improve mixing efficiency, researchers explored multiple paths, each expanding the algorithm toolbox and optimizing convergence behavior both theoretically and empirically. Gradient-based methods, such as Hamiltonian Monte Carlo (HMC), were among them. [4] Its adaptive extension No-U-Turn Sampler (NUTS) [5] By leveraging Hamiltonian dynamics to generate long-range proposals, significant improvements are achieved in high-dimensional settings with high acceptance rates. HMC simulations of Hamiltonian trajectories avoid local traps in random walks, while NUTS further reduces manual parameter tuning through adaptive path length selection. Subsequent improvements include splitting HMC to reduce computational overhead. [6] and Stochastic Gradient HMC to adapt to large-scale datasets [7] Together, they demonstrated that dynamically guided transfer can significantly reduce the integral autocorrelation time (IACT) and exhibits superior scalability in applications such as Bayesian neural networks and large-scale image analysis. [8][9] For example, stochastic variants approximate full gradient computation on large datasets, balancing accuracy and efficiency.
[0006] Parallel research lines employ an operator theory perspective, accelerating convergence by widening the spectral gap or attenuating slow eigenmodes. This spectral viewpoint emphasizes the crucial role of operator eigenstructure in determining the mixing rate. Techniques such as irreversible perturbations...
[10] Deterministic jump
[11] and local information suggestions
[12]
[13] Efficient proofs are provided in both discrete and continuous state spaces. These advances are theoretically based on the Kipnis–Varadhan central limit theorem.
[14] This relates the spectral structure to the asymptotic variance of the ergodic average. Recent extensions incorporate discrete spatial gradient MCMC enhancements, such as continuous relaxation or Wasserstein gradient flow, to achieve efficient sampling of discrete targets.
[15]
[16] This provides a unified framework for hybrid state spaces. Furthermore, randomization techniques and local approximation have been used to accelerate MCMC in inverse problems and function sampling, alleviating the computational burden of full data dependencies.
[17]
[18] .
[0007] Based on these developments, deterministic fixed-point acceleration schemes have recently been integrated into MCMC design, bridging optimization theory and random sampling to foster new algorithmic paradigms. Classical iterative approaches such as Mann...
[19] and Ishikawa
[20] Prove that multi-step convex combinations can accelerate convergence without changing the positions of fixed points.
[19]
[20]
[21]
[22] These iterations progressively refine historical updates to suppress oscillatory behavior and ensure a faster approach to the fixed point. Anderson acceleration achieves significant speedups in nonlinear problems by extending this principle through linear extrapolation of historical iterations.
[23]
[24]
[25] The unifying theme is that multi-step operator transformations suppress dominant slow modes, providing a promising mechanism for accelerating Markov chains while maintaining invariance. In the context of MCMC (Multi-Step Machine Learning), this paradigm has been preliminarily applied to accelerate value iteration in reinforcement learning.
[26] Solving for non-smooth fixed points
[27] This highlights its potential in stochastic environments. Although these efforts are primarily aimed at deterministic optimization, their core mechanism—multi-step mixing to suppress slow modes—provides transferable insights into the spectral refinement of MCMC transfer kernels.
[0008] Based on this, this paper proposes an improved MCMC algorithm based on Ishikawa iteration. By constructing a multi-step mixing operator on the MH transfer kernel, the new transfer operator effectively compresses non-trivial eigenvalues while maintaining the target distribution π as an invariant measure, thereby accelerating chain mixing. The core of this algorithm lies in incorporating multi-step convex combination into the transfer process, ensuring the widening of the spectral gap without introducing bias. Through Hilbert spatial spectral decomposition and Kipnis–Varadhan theory, we prove that this method can significantly reduce IACT and asymptotic sample variance.
[0009] In view of the shortcomings of the prior art described above, the purpose of this invention is to provide a polynomial Markov operator construction method based on Ishikawa iteration, which solves the problems of small gaps in the spectrum of transfer operators, slow convergence speed and high asymptotic variance in the existing Markov chain Monte Carlo method.
[0010] To achieve the above and other related objectives, this invention provides a method for constructing a polynomial Markov operator based on Ishikawa iteration, characterized in that: by introducing an Ishikawa two-stage iterative structure, a polynomial Markov operator with improved spectral properties is obtained, and the method consists of four parts.
[0011] The first part is: Construction of basic Markov operators and definition of function space.
[0012] In measurable space Above, given the target probability distribution Constructing to satisfy Invariant Markov transfer nuclei And define it in Hilbert space. The function of the above is:
[0013]
[0014] Where the function This operator is a self-adjoint contraction operator under the condition of invertibility, and its spectral properties determine the convergence behavior of the Markov chain.
[0015] The second part is: the introduction of the Ishikawa iterative structure.
[0016] Based on the Ishikawa iterative framework of fixed-point theory
[0017]
[0018] Where the parameter sequence .
[0019] Introducing a two-stage operator construction mechanism for any function Define intermediate operators:
[0020]
[0021] in The quadratic composite operator representing the transfer kernel, with parameters... This structure enhances the global exploration capability of the operator by introducing secondary transition information.
[0022] The third part is: the construction of polynomial Markov operators.
[0023] Based on the aforementioned intermediate operator, parameters are introduced. Construct the final operator:
[0024]
[0025] Right now:
[0026]
[0027] This operator is a convex combination of the identity operator, the first-order operator, and the second-order operator, forming a polynomial Markov operator.
[0028] The fourth part is: Operator property analysis and performance improvement mechanism.
[0029] Based on spectral theory analysis, this construction method has the following properties:
[0030] Invariance preservation: if ,but ;
[0031] (2) Self-adjointness: When When reversible, For self-adjoint operators;
[0032] (3) Spectral transform properties:
[0033]
[0034] (4) Spectral gap enhancement: The operator has a larger spectral gap than the original operator, thereby improving the convergence speed;
[0035] (5) Asymptotic variance decreases: for any The function, wherein the asymptotic variance corresponding to the operator is not greater than that of the original operator.
[0036] As described above, the present invention proposes a polynomial Markov operator construction method based on Ishikawa iteration. By introducing a higher-order transition structure and a two-stage iteration mechanism, it effectively improves the spectral properties of Markov chains and increases sampling efficiency while keeping the target distribution unchanged. This provides a new operator design method for high-dimensional Bayesian inference and complex distribution sampling. Attached Figure Description
[0038] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort.
[0039] Figure 1 This is a flowchart illustrating the construction of a polynomial Markov transition operator based on Ishikawa iteration, as an example of the present invention. Detailed Implementation
[0040] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention. These all fall within the scope of protection of the present invention.
[0041] refer to Figure 1 The purpose of this invention is to provide a method for constructing polynomial Markov operators based on Ishikawa iteration, comprising the following steps:
[0042] Construction of basic Markov operators and definition of function space:
[0043] set up For measurable space, The target probability measure on it is called a mapping.
[0044]
[0045] For Markov transfer kernels, that is, for any ,function Measurable, and for any , It is a probability measure.
[0046] For probability measure With measurable functions ,definition:
[0047]
[0048]
[0049]
[0050] set up for Invariant measure, i.e.
[0051]
[0052] Then for any It has shrinkage properties
[0053]
[0054] thereby It can be regarded as acting on Bounded linear operators on.
[0055] If reversibility (detailed equilibrium condition) is further satisfied:
[0056]
[0057] but exist The above is a self-adjoint contraction operator, whose spectral properties determine the convergence behavior of the Markov chain.
[0058] for ,make Define the centralization function , 0, and record the lag. covariance is If the following summation converges, the autocorrelation function is: Autocorrelation Time (IACT) is defined as follows:
[0059]
[0060] Introduction of the Ishikawa iterative structure:
[0061] set up For the actual Banach space, For a nonempty closed convex set, the mapping It is a non-extension operator, that is, it satisfies
[0062]
[0063] Ishikawa proposed the following two-step iterative format:
[0064]
[0065] Where the parameter sequence .
[0066] Structurally, this iteration consists of two convex combinations, and its essence can be expressed as the following composite mapping process:
[0067]
[0068] This iteration essentially constructs a "look-ahead two-step update mechanism," which introduces additional operator information compared to single-step iteration.
[0069] Construction of polynomial Markov operators:
[0070] To apply the above iterative theory to the MCMC framework, the following correspondence is established as shown in Table 1:
[0071] Fixed-point theoretical framework correspondence table
[0072] Fixed point theory Markov chain framework Banach Space Hilbert Space Non-extended mapping Markov operators fixed point Invariant function Non-expansive Shrinkage
[0073] Furthermore, under the assumption about When invertible (i.e., satisfying the detailed balance condition), the operator exist The above is a self-adjoint operator, thus possessing a spectral decomposition structure.
[0074] Within this framework, the following basic conclusions can be drawn:
[0075] 1. Non-extensibility of Markov operators
[0076] set up For For a Markov operator with an invariant distribution, then for any ,have
[0077]
[0078] prove
[0079] From Jensen's inequality and the contractility of conditional expectation:
[0080]
[0081] right about Points, using ,get:
[0082]
[0083] 2. Fixed-point structure
[0084] Under irreducible conditions, if ,but for - Almost everywhere a constant.
[0085] therefore:
[0086]
[0087] Target distribution This corresponds to the fixed-point structure.
[0088] In the operator framework, such as Figure 1 Below, the Ishikawa iteration is written as:
[0089] first step:
[0090]
[0091] Step Two:
[0092]
[0093] Substitution
[0094]
[0095] Expand
[0096]
[0097] Therefore, the following operator expression is obtained:
[0098]
[0099] The operator is It is a quadratic polynomial function, and therefore can be called an Ishikawa-type polynomial Markov operator.
[0100] Operator property analysis and performance improvement mechanisms:
[0101] In this section, we establish the spectral acceleration and variance improvement properties of Ishikawa-type operators in general state space relative to the original Markov operators. The core idea is:
[0102]
[0103] As a polynomial spectral transform, it "compresses" the spectrum in the slow-mode region, thereby improving mixing efficiency.
[0104] set up To act on The self-adjoint Markov operator.
[0105] Define its spectral gap as
[0106]
[0107] Equivalent land,
[0108]
[0109] 1. Spectral gap enhancement
[0110] set up Let be an invertible Markov operator whose spectrum is contained in [−1, 1]. Definition
[0111]
[0112] And record
[0113]
[0114] Then we have:
[0115] The spectrum satisfies the mapping relationship
[0116]
[0117] in
[0118]
[0119] like for The second largest eigenvalue, then
[0120]
[0121] like ,but
[0122]
[0123] prove
[0124] By the spectral theorem:
[0125]
[0126] Therefore, the pointwise mapping of the spectrum holds true.
[0127] right Expand:
[0128]
[0129] then
[0130]
[0131] The conclusion is to take the maximum value.
[0132] 2. Spectral representation of asymptotic variance
[0133] right The asymptotic variance can be expressed as (Kipnis–Varadhan formula):
[0134]
[0135] Similarly,
[0136]
[0137] 3. Asymptotic variance monotonicity under the condition of increased paving gap, if it satisfies
[0138]
[0139] Then for any ,have
[0140]
[0141] prove
[0142] function
[0143]
[0144] It is monotonically increasing on (−1,1)(-1,1)(−1,1).
[0145] like ,but
[0146]
[0147] Integrating over the spectral measure yields:
[0148]
[0149] annotation:
[0150] The above conditions are equivalent to:
[0151]
[0152] Therefore in The upper bound holds. That is, when the spectrum is mainly concentrated in the non-negative region, the variance improvement holds.
[0153] We directly prove at the level of the transfer nucleus It retains reversibility and explains its structure.
[0154] 4. Core layer structure and reversibility
[0155] set up It is a reversible nucleus, that is
[0156]
[0157] Define a new core:
[0158]
[0159] but:
[0160] It is a Markov nucleus;
[0161] about reversible;
[0162] The operator form is as follows: .
[0163] prove
[0164] (1) Probabilistic
[0165] Obviously:
[0166]
[0167] (2) Reversibility
[0168] That is, verification:
[0169]
[0170] Due to the identity term
[0171]
[0172] Established.
[0173] The second item (one-step verification)
[0174] Depend on Reversible:
[0175]
[0176] The third item (two-step verification)
[0177] consider:
[0178]
[0179] By changing the order of integration, we get
[0180]
[0181] Utilizing reversibility:
[0182]
[0183] get:
[0184]
[0185] Symmetry gives:
[0186]
[0187] Therefore, the overall condition of fine balance is met.
[0188] (3) Operator consistency
[0189] For any :
[0190]
[0191] Right now:
[0192]
[0193] Concluding summary:
[0194] (1). The Ishikawa-type construction is essentially a spectral polynomial acceleration of Markov operators;
[0195] (2). In the slow mode region, its implementation
[0196]
[0197] This improves the spectral gap;
[0198] (3). In the typical (non-negative spectrum) case, it guarantees
[0199]
[0200] That is, asymptotic variance improvement;
[0201] (4). Moreover, this improvement maintains complete reversibility and feasibility in the transfer core layer.
[0202] In conclusion, based on theoretical analysis and research results on the spectral properties of operators, this patent proposes a polynomial Markov operator construction method based on Ishikawa iteration. This method effectively improves the spectral structure of traditional Markov transition operators while maintaining the target distribution unchanged. Compared with single transition operators or existing linear combination methods, this method introduces a quadratic composite transition structure and a two-stage iteration mechanism, thereby expanding the spectral gap of the operator, improving the convergence speed of the Markov chain, and reducing the asymptotic variance of the corresponding statistics. Simultaneously, this method provides a new operator design framework for Markov chain Monte Carlo algorithms, which can be widely applied in fields such as high-dimensional probability distribution sampling, Bayesian statistical inference, and complex system simulation. Furthermore, the polynomial operator construction concept established in this invention also provides a theoretical foundation for the further development of higher-order Markov operators and their spectral optimization methods.
[0203] The above embodiments should be understood as illustrative only and not as limiting the scope of protection of the present invention. After reading the description of the present invention, those skilled in the art can make various alterations or modifications to the present invention, and these equivalent changes and modifications also fall within the scope defined by the claims of the present invention.
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Claims
1. A method for constructing a polynomial Markov operator based on Ishikawa iteration, characterized in that: Applied to the Markov chain Monte Carlo method, by introducing the Ishikawa two-stage iterative structure, the basic Markov transition operator is extended to a higher order. The identity operator, the first-order transition operator and its second-order composite operator are constructed by convex combination, thereby obtaining a polynomial Markov operator with improved spectral properties, which is used to improve the convergence speed of the Markov chain and reduce the asymptotic variance.
2. The method for constructing a polynomial Markov operator based on Ishikawa iteration as described in claim 1, characterized in that: In measurable space Above, given the target probability distribution Constructing to satisfy Invariant Markov transfer nuclei And define it in Hilbert space. The function of above is 3. The method for constructing a polynomial Markov operator based on Ishikawa iteration as described in claim 2, characterized in that: The Markov operator Satisfying the reversibility condition, thus in Spatially, it is a self-adjoint contraction operator.
4. The method for constructing a polynomial Markov operator based on Ishikawa iteration as described in claim 3, characterized in that: Based on the Ishikawa iterative structure, an intermediate operator is constructed: Where parameters ,and This represents a quadratic composite operator for the transfer kernel.
5. The method for constructing a polynomial Markov operator based on Ishikawa iteration as described in claim 4, characterized in that: Based on the intermediate operator, parameters are introduced. Construct the final operator: ( Right now:
6. The method for constructing a polynomial Markov operator based on Ishikawa iteration as described in claim 5, characterized in that: The operator It satisfies the invariance preservation property, that is:
7. The method for constructing a polynomial Markov operator based on Ishikawa iteration as described in claim 6, characterized in that: When the Markov operator When reversible, the operator exist It is a self-adjoint operator in space.
8. The method for constructing a polynomial Markov operator based on Ishikawa iteration as described in claim 7, characterized in that: The operator The spectrum is given by the following function mapping:
9. The method for constructing a polynomial Markov operator based on Ishikawa iteration as described in claim 8, characterized in that: The operator compared to the original operator It has a larger spectral gap, thereby improving the convergence speed of the Markov chain.
10. The method for constructing a polynomial Markov operator based on Ishikawa iteration as described in claim 9, characterized in that: For any square-integrable function, the asymptotic variance corresponding to the operator is no greater than that of the original Markov operator, thereby improving the estimation accuracy.