The Geometry of Information from Shannon Channel Capacity to Artificial Intelligence
Shannon channel capacity can be interpreted through the geometry of signals, noise, and distinguishable regions in high-dimensional spaces. This perspective provides a shared framework for the limited-capacity and robustness-to-distortion problems of embedding and latent spaces in representation learning.
C = B × log₂(1 + S/N)
This formula determines the maximum information rate that can be transmitted reliably through a bandwidth-limited and noisy channel. Here, C denotes channel capacity, B bandwidth, and S/N the signal-to-noise ratio. The scope of information theory is not limited to communication systems; learned communication, data compression, the information bottleneck, and some representation-learning studies also use theoretical tools such as entropy, mutual information, and rate-distortion.
A geometric interpretation of channel capacity
Channel capacity can also be explained geometrically. In his 1961 book An Introduction to Information Theory: Symbols, Signals and Noise, J. R. Pierce addresses information theory through high-dimensional spaces. In this approach, a signal measured over a given interval is regarded as a high-dimensional vector. The signal energy determines the vector length, while noise determines the uncertainty region surrounding that vector.
Under appropriate stationarity conditions, long signal and noise sequences concentrate on particular energy shells in high-dimensional space. This behavior does not directly accord with intuition derived from three-dimensional space. Nevertheless, it is one of the fundamental elements that makes a geometric explanation of channel capacity possible.
A high-dimensional sphere-packing problem
In the geometric interpretation, channel capacity is reduced to the following question: How many distinct noise regions can be placed inside the large volume formed by signal and noise without overlapping?
This is a high-dimensional sphere-packing problem. Each possible transmission can be associated with a region in signal space; when the uncertainty regions produced by noise remain sufficiently separate, the receiver can reliably distinguish the transmitted signal. The number of separable regions that can be placed is related to the number of distinct messages that can be transmitted reliably.
A shared geometric framework with representation learning
A similar geometric problem arises in artificial intelligence. Many representation-learning models represent data in a high-dimensional embedding or latent space. These representations are expected to remain distinguishable under data variation, noise, and other perturbing effects.
Not every artificial intelligence model uses the Shannon channel-capacity formula directly. The relationship is not a one-to-one mathematical equivalence, but rather a shared geometric perspective grounded in information theory. The question Shannon posed in data communication reappears in a different form in artificial intelligence: How much information can be placed in a limited representation space while preserving its meaning and remaining distinguishable despite perturbing effects?
