…as a sentient being.
Fourier Transforms & the Wave Packet
Fourier analysis: In quantum mechanics, a particle’s wave function can be decomposed into its momentum components using Fourier transforms.
The relationship between position & momentum uncertainties can be interpreted using the Fourier transform, which describes the degree of spreading of a wave in both position and momentum space.
Wave packet:
A localised particle or superposition of multiple wave functions; typically sine or cosine waves of different frequencies, is represented by a wave packet, which has a spread in both position & momentum space.
A narrower wave packet in position implies a broader distribution in momentum & vice versa.
Mathematically, a wave packet is often expressed as:
where A(k) is the amplitude, k is the wave number & w is the angular frequency.
An example of Fourier Transformations & Wave Packets used to analyse Knowledge Graphs from Neural Networks in order to train AI & set system States that influences ai output
Analysing knowledge graphs using Fourier transformations & wave packets is an advanced approach that can bring new insights to AI training & system states. Here’s a breakdown of how you might approach this idea:
1. Knowledge Graphs in Neural Networks
Knowledge Graphs represent information as a graph structure where entities are nodes & relationships are edges. These are often used in AI systems to represent facts, concepts, and their interconnections.
It’s often represented as a graph ( G = (V, E) ), where ( V ) is a set of vertices (representing concepts or entities) and ( E ) is a set of edges (representing relationships)
Neural networks can learn from these graphs by translating them into vector space, using methods like Graph Neural Networks or embeddings like TransE, Node2Vec, or GraphSAGE.
2. Fourier Transformations in Knowledge Graphs
The Fourier Transform is a tool to convert a signal from its time domain to its frequency domain. In the context of graphs, the graph Laplacian is analogous to the Laplacian operator in signal processing, which allows for Spectral Graph Theory.
Graph Fourier Transform can be applied to knowledge graphs by treating the graph as a signal. The GFT decomposes the graph into its frequency components, identifying smooth variations over the graph structure and highlighting periodicity, clusters, or patterns.
High frequency components can indicate rapid changes (e.g., weak relationships or outliers).
Low frequency components can show strong, stable relationships or clusters.
3. Wave Packets & Graph Analysis
Mapping Knowledge Graph to a Wave Packet:
In this analogy:
Nodes correspond to frequencies or wave numbers.
Each node represents a wave with certain properties (e.g., amplitude, frequency, phase) related to the entity’s importance or characteristics in the knowledge graph.
Edges (Relationships) are like interference patterns between waves.
Interactions between nodes/entities create constructive or destructive interference, representing how different entities or concepts interact or reinforce each other.
Suppose we have ( N ) nodes in the knowledge graph.
Each node ( ni ) can be associated with a wave function
where:
Ai : amplitude representing the importance of the node.
ki: wave number representing the conceptual position of the node in knowledge space.
wi : frequency representing the relevance or time-dependent behaviour of the concept.
The overall knowledge wave packet
can be expressed as the superposition of these waves:
Here, the sum of the individual node-waves creates a wave packet, where interactions between nodes (i.e., relationships in the graph) determine the constructive or destructive interference, shaping the form of the packet.
Interpreting
Localised Knowledge:
In a similar way that a wave packet is localised in space due to interference patterns, the knowledge graph wave packet represents localised knowledge or meaning.
For example, clusters of nodes with strong relationships would correspond to areas of high amplitude in the wave packet.
In the context of knowledge graphs, a wave packet could represent a localised feature or cluster of entities in the graph.
Using wavelets allows for multiscale analysis of the graph, enabling examination at different levels of granularity. Graph Wavelets can be used to identify both global & local patterns within knowledge graphs.
This multiscale approach enables the detection of both large, overarching trends (global structures) & smaller, detailed substructures (local features) within the graph.
Evolution of Knowledge: The time evolution of the wave packet (wt) can represent the changing relevance or importance of different entities over time.
Example:
Imagine a knowledge graph with three nodes:
A , B & C with relationships between A & B as well as between B and C.
Each node corresponds to a wave & the interference patterns from their connections create a wave packet.
If the relationships between these nodes are strong, you might see constructive interference (higher peaks), while weaker relationships lead to more spread out, less defined regions (lower amplitude areas).
Visual Representation:
The resulting wave packet can be visualised as a complex, dynamic , oscillating structure. Peaks signify regions where entities are densely clustered or strongly related, while troughs indicate areas where connections are sparse or weaker, highlighting variations in the underlying structure.
Expressing a knowledge graph as a wave packet captures the dynamic & interconnected nature of knowledge, using the principles of wave interference and superposition.
Training AI Models with Spectral Features
Spectral features from GFT & wavelet transforms can be used as input to AI models, enabling the neural network to learn from both the spatial structure & frequency components of the knowledge graph.
For example, you could train a GNN or other neural models using these spectral embeddings, allowing the AI to capture both topological and spectral properties of the data.
Frequency domain features (low/high-frequency components) can serve as a feature set for predicting relationships or node embeddings within the graph, refining the system’s ability to generalise and identify patterns.
Setting System States with Fourier & Wavelet Data
Once you have analysed a knowledge graph using Fourier & wavelet techniques, the spectral data can be used to set system states or influence the AI’s behaviour:
State Representation:
Use the spectral features of the graph to set the initial states of the system. For example, initialise the neural network based on the frequency domain representation of the input knowledge graph.
Feedback Mechanism:
During training, use changes in the graph’s spectral components to adjust or update the system states dynamically. This could allow for adaptive learning, where the system fine tunes its internal states based on how relationships in the knowledge graph evolve.
Regularisation & Constraints:
Spectral information could also act as a form of regularisation, guiding the network away from overfitting by enforcing smoothness or periodicity constraints in the learned graph structure.
Influencing AI Output
By integrating spectral information from Fourier or wavelet transformations into the neural network’s latent space (e.g., the embeddings), the system’s output can be fine tuned, where as a mechanism, it is used for setting system states.
For instance, high-frequency components can be used to detect anomalies or outlier relationships in the graph, which can then be flagged in AI outputs (e.g., knowledge inference, recommendations).
Low frequency components can influence long-term patterns or trends the AI should emphasise, leading to more stable & robust outputs.
It also provides a mechanism for setting system states, which can guide AI outputs in a more informed and stable manner.
Comments