LSA / LSI / LDA Geometry

This guide records the platform's intended interpretation of LSA, LSI, and LDA.

Integer substrate

The semantic stack begins on a discrete substrate: characters, tokens, counts, and document-term structure.

We can represent the corpus as a weighted matrix:

$$A\in {\mathbb{R}}^{m\times n}$$

where $m$ is the vocabulary dimension and $n$ is the document dimension.

LSA as a closed latent basis

For LSA we compute a truncated singular value decomposition:

$$A\approx U_k{\mathrm{\Sigma }}_k{V}_k^{\mathrm{\top }}$$

In this platform's conceptual model, LSA is the closed latent basis. Once $k$ is fixed, we are working inside a bounded latent vector regime.

This makes LSA well suited for:

• deterministic linear-algebraic compression
• similarity
• clustering
• reconstruction-aware semantic analysis

LSI as the bridge regime

In this platform's model, LSI is not identical to LSA in role, even though both depend on truncated SVD ideas.

LSI is the bridge between local closed structure and broader search/retrieval dynamics. It projects queries and documents into the latent space for operational retrieval:

$$q^{\prime }=q^{\mathrm{\top }}{U}_k{\mathrm{\Sigma }}_k^{-1}$$

and document embeddings can be represented in the latent basis by:

$$d_i={\mathrm{\Sigma }}_k{V}_{k,i}^{\mathrm{\top }}$$

So in our conceptual language:

• LSA = closed basis
• LSI = curved bridge between closed basis and broader retrieval space

LDA as the open topic family

For LDA, documents are mixtures of topics and topics are distributions over words:

$${\theta }_d\sim \mathrm{Dir}(\alpha )$$$${\varphi }_k\sim \mathrm{Dir}(\eta )$$$$z_{d,n}\sim \mathrm{Cat}({\theta }_d),w_{d,n}\sim \mathrm{Cat}({\varphi }_{z_{d,n}})$$

In this platform's conceptual model, LDA is the open topic family. It is the unbounded topic simplex beyond the fixed closed basis regime.

Projection, annealing, diffusion

The diagrams imply three directional transformations:

Projection

Move from discrete observed substrate into latent structure.

Annealing

Smooth local noisy structure into stable closed latent basis.

Diffusion

Open the closed latent basis outward into broader topic families and semantic drift.

The aligned 23rd channels

The 22-basis regime remains the contributing latent family. The comparison happens at the 23rd channel:

$$c_{23}^{LSA}=f_{23}(U_{22},{\mathrm{\Sigma }}_{22},V_{22})$$$$c_{23}^{LSI}=g_{23}(q,U_{22},{\mathrm{\Sigma }}_{22},V_{22})$$$$c_{23}^{LDA}=h_{23}(\theta ,\varphi ,\alpha ,\eta )$$

These are aligned variants:

• LSA23 = closed summary channel over the latent LSA-22 basis
• LSI23 = bridge channel from local closed structure toward broader retrieval drift
• LDA23 = open topic extension beyond the fixed latent basis family