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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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Cluster Sampling

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Dr. Rishabh Pathak

Cluster sampling is a sampling technique used in research and statistical studies where the population is divided into groups or clusters, and a random sample of these clusters is selected for analysis. Instead of individually selecting elements from the population, cluster sampling involves selecting entire groups or clusters and then sampling within those selected clusters. Here are some key points about cluster sampling:


1.    Definition:

o    In cluster sampling, the population is divided into clusters or groups based on certain characteristics (geographic location, organizational units, etc.). A random sample of clusters is then selected, and data is collected from all elements within the chosen clusters.

2.    Process:

o    The steps involved in cluster sampling include:

§  Dividing the population into clusters.

§  Randomly selecting a sample of clusters.

§  Collecting data from all elements within the selected clusters.

§  Analyzing the data to draw conclusions about the entire population.

3.    Advantages:

o    Cluster sampling is often more cost-effective and practical than other sampling methods, especially when the population is large and widely dispersed. It can reduce the time and resources required for data collection by focusing on selected clusters rather than individual elements.

4.    Disadvantages:

o    One potential drawback of cluster sampling is the risk of increased sampling error compared to other sampling methods like simple random sampling. Variability within clusters can affect the precision of estimates, especially if clusters are not homogeneous.

5.    Examples:

o    An example of cluster sampling is conducting a survey in a city by dividing the city into neighborhoods (clusters) and randomly selecting a sample of neighborhoods. Data is then collected from all households within the selected neighborhoods to represent the entire city population.

6.    Types:

o    There are different types of cluster sampling, including:

§  Single-stage cluster sampling: Where clusters are selected and all elements within the chosen clusters are included in the sample.

§  Multi-stage cluster sampling: Where clusters are selected in stages, with further sampling within selected clusters to obtain the final sample.

7.    Applications:

o    Cluster sampling is commonly used in fields such as public health, sociology, market research, and environmental studies. It is particularly useful when it is impractical to sample individuals directly or when the population is naturally grouped into clusters.

8.    Considerations:

o  When using cluster sampling, researchers should ensure that clusters are representative of the population and that the sampling process within clusters is random to maintain the validity and generalizability of the study results.

Cluster sampling offers a practical and efficient way to obtain representative samples from large and diverse populations, making it a valuable tool in various research contexts. By carefully designing the sampling process and addressing potential sources of bias, researchers can leverage cluster sampling to make reliable inferences about the target population.

 

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