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Energy Manifold Natural Gradient Descent: From Riemannian Optimization to Modern Neuroscience, NeuroAI and Quantum Physics

Dr. Rishabh Pathak
When Geometry, Energy, Artificial Intelligence and Neuroscience Converge Modern Artificial Intelligence is rapidly moving beyond the idea that learning simply means minimizing an error function. Increasingly, researchers are asking a deeper question: what is the structure of the space in which learning takes place? This question becomes particularly important when the system being modelled is constrained, nonlinear, dynamic, or governed by physical principles. A recent work titled “Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers” , by Zhangyong Liang and Huanhuan Gao, introduces Energy Manifold Natural Gradient Descent (EMNGD) , a mathematical framework that extends energy-based natural-gradient optimization from unconstrained Euclidean parameter spaces to constrained Riemannian parameter manifolds . At its core, the framework proposes a simple but powerful principle: An optimization algorithm should not only determine how to reduce error; it sh...
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Complex Random Sampling

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Dr. Rishabh Pathak

Complex random sampling designs involve the use of mixed sampling procedures that combine elements of probability and non-probability sampling techniques. These designs are employed in research studies where simple random sampling may not be feasible or where specific sampling requirements need to be met. Here are some key points about complex random sampling:


1.    Mixed Sampling Procedures:

o    Complex random sampling designs often combine elements of probability sampling (random selection) with non-probability sampling techniques to achieve specific sampling goals. These designs may involve stratification, clustering, or systematic sampling methods.

2.    Systematic Sampling:

o    Systematic sampling is a common type of complex random sampling where every ith element in a list is selected after a random start. This method introduces an element of randomness by selecting a random starting point, but subsequent selections follow a fixed interval pattern. Systematic sampling can be more efficient than simple random sampling in certain situations.

3.    Randomness and Even Distribution:

o    Complex random sampling designs aim to achieve randomness in sample selection while ensuring an even distribution of sample elements across the population. This helps in reducing bias and increasing the representativeness of the sample.

4.    Efficiency and Cost-Effectiveness:

o    While complex random sampling designs may involve more intricate sampling procedures than simple random sampling, they can offer advantages in terms of efficiency and cost-effectiveness, especially when dealing with large populations or specific sampling constraints.

5.    Representativeness and Generalizability:

o    The goal of complex random sampling designs is to create samples that are representative of the population and can be generalized to make inferences about the larger target population. By incorporating elements of randomness and structure, these designs aim to enhance the validity of research findings.

6.    Statistical Rigor:

o    Complex random sampling designs require careful planning and implementation to ensure statistical rigor in the sampling process. Researchers must consider factors such as sample size, sampling intervals, and stratification criteria to achieve valid and reliable results.

7.    Application in Research:

o    Complex random sampling designs are commonly used in survey research, epidemiological studies, and other research contexts where simple random sampling may not be sufficient to address the research objectives. These designs offer flexibility and customization in sample selection.

Complex random sampling designs play a crucial role in research methodology by providing researchers with options to tailor their sampling strategies to specific research needs. By combining elements of randomness and structure, these designs aim to produce samples that are both representative of the population and suitable for making valid inferences.

 

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