Approximate Fiber Product: A Preliminary Algebraic-Geometric Perspective on Multimodal Embedding Alignment

Each image is divided into patches, where each patch consists of discrete pixel intensity values in the range $[0,255]$. By flattening the pixel values of a patch into a vector $(a_{0},a_{1},\dots,a_{n})$, we construct the corresponding polynomial:

This representation allows the image patches to be viewed as elements in the polynomial ring $\mathbb{Z}_{256}[x]$, enabling algebraic manipulation and analysis.

For text, the input is tokenized into a sequence of token IDs $(t_{0},t_{1},\dots,t_{m})$, where each token $t_{i}$ is an integer in the range $[0,|V|-1]$, and $|V|$ is the vocabulary size. The corresponding polynomial representation is:

This representation embeds the discrete token sequences into the polynomial ring $\mathbb{Z}_{|V|}[x]$, capturing their inherent sequential structure.

By representing images and text as polynomials in their respective rings $\mathbb{Z}_{256}[x]$ and $\mathbb{Z}_{|V|}[x]$, we provide a common algebraic framework for multimodal data. These polynomial representations serve as a foundation for introducing algebraic geometry tools, such as fiber products and moduli spaces, to study the alignment and structure of multimodal embeddings.

The size of the approximate fiber product is given by:

where $B_{\epsilon}(z)=\{z^{\prime}\in Z\mid\|z-z^{\prime}\|\leq\epsilon\}$ represents an $\epsilon$-neighborhood around $z$. This relationship reveals that $|X\times_{Z,\epsilon}Y|$ depends on the overlap of $\mu_{f}$ and $\mu_{g}$. For high-density overlap regions, the growth of $|X\times_{Z,\epsilon}Y|$ with $\epsilon$ is rapid, while minimal overlap results in slower growth.

When $\mu_{f}$ and $\mu_{g}$ are Gaussian distributions in $d$-dimensional space, the size of the approximate fiber product asymptotically scales as:

Here, $\epsilon^{d}$ reflects the dependency on the dimensionality $d$, and $\|\mu_{f}-\mu_{g}\|$ determines the effective overlap. This result emphasizes that higher dimensions require careful tuning of $\epsilon$ to maintain alignment precision.

To evaluate robustness, consider the perturbed embeddings $f_{\delta}(x)=f(x)+\delta_{f}(x)$ and $g_{\delta}(y)=g(y)+\delta_{g}(y)$, where $\delta_{f}(x)$ and $\delta_{g}(y)$ are bounded noise terms ($\|\delta_{f}(x)\|,\|\delta_{g}(y)\|\leq\eta$). The approximate fiber product satisfies the inclusion:

if and only if $\eta\leq\epsilon/2$. This condition ensures that the alignment is robust to bounded noise, providing stability in noisy embedding spaces.

The effective dimensionality of the alignment region is determined by:

where $Z_{s}$ is the shared semantic subspace in $Z$. This highlights the importance of embedding both modalities into well-structured subspaces, minimizing dimensional redundancy and maximizing overlap.

Finally, the parameter $\epsilon$ controls the size and flexibility of the alignment region:

Choosing $\epsilon$ optimally involves balancing precision and flexibility, ensuring meaningful alignment while accounting for noise.

This result ensures that the approximate fiber product inherits compactness from the embedding space $Z$, facilitating numerical computations and stability analysis.

This theorem formalizes the behavior of $|X\times_{Z,\epsilon}Y|$ under extreme values of $\epsilon$, providing a theoretical foundation for alignment size analysis.

Let $\Pi_{s}$, $\Pi_{I}$, and $\Pi_{T}$ denote the orthogonal projection operators onto $Z_{s}$, $Z_{I}$, and $Z_{T}$, respectively. For any $z\in Z$, the decomposition can be written as:

The projection operators satisfy the following properties:

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  Orthogonality: $\Pi_{s}\cdot\Pi_{I}=\Pi_{s}\cdot\Pi_{T}=\Pi_{I}\cdot\Pi_{T}=0$.

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  Completeness: $\Pi_{s}+\Pi_{I}+\Pi_{T}=\mathrm{Id}_{Z}$, where $\mathrm{Id}_{Z}$ is the identity operator on $Z$.

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  Preservation: For $z\in Z_{s}$, $Z_{I}$, or $Z_{T}$, the corresponding projection is the identity, e.g., $\Pi_{s}(z_{s})=z_{s}$.

Let $f:I\to Z$ and $g:T\to Z$ be the embedding functions for images and text, respectively. The embeddings can also be decomposed into their subspace components:

where:

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  $f_{s}(i),g_{s}(t)\in Z_{s}$: Represent the shared semantic components in the shared subspace.

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  $f_{I}(i)\in Z_{I}$: Represents the modality-specific component for images.

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  $g_{T}(t)\in Z_{T}$: Represents the modality-specific component for text.

By doing so, the decomposition ensures that shared and modality-specific properties are considered.

Embedding space decomposition can be understood through the lens of sheaf theory. Consider the shared embedding space $Z$ decomposed into open subsets $Z_{s},Z_{I},Z_{T}$, representing shared, image-specific, and text-specific subspaces, respectively. Define a presheaf $\mathcal{F}$ over $Z$ such that for each open set $U\subset Z$, $\mathcal{F}(U)$ captures the set of embeddings consistent with $U$.

To ensure the alignment of local embeddings with the global decomposition, $\mathcal{F}$ must satisfy the sheaf condition:

where $\{U_{i}\}$ is an open cover of $U$. This sheaf-theoretic perspective formalizes the compatibility of local embeddings with the global structure of $Z$, ensuring consistency between shared and modality-specific features.

The shared semantic subspace $Z_{s}$ can be modeled as an algebraic variety embedded in the larger space $Z$. For instance, $Z_{s}$ might be represented as the solution set of a system of polynomial equations:

where $P_{i}$ are polynomials over $Z$. This algebraic structure provides additional constraints on embeddings, ensuring that shared features align along well-defined geometric loci.

Given the fiber product construction:

the shared space $Z_{s}$ acts as a base variety, and the alignment condition enforces that the projections $f(i)$ and $g(t)$ lie in a tubular neighborhood around $Z_{s}$. This geometric constraint simplifies the analysis of alignment stability and efficiency.

For any $z\in Z$, its decomposition $z=z_{s}+z_{I}+z_{T}$ ensures that the projection operators $\Pi_{s}$, $\Pi_{I}$, and $\Pi_{T}$ satisfy:

This partitioning provides a quantitative measure of how embeddings distribute their information across the shared and modality-specific subspaces.

The shared semantic subspace $Z_{s}$ acts as the intersection of the image and text embedding distributions. Formally:

The dimensionality of $Z_{s}$ determines the capacity of the shared space to capture common features. If the projections are linearly dependent, $\dim(Z_{s})$ will shrink, limiting alignment capacity.

The quality of alignment depends on the degree of overlap between $Z_{s}$, $Z_{I}$, and $Z_{T}$. Consider the alignment error:

where $\lambda$ controls the penalty for misalignment. Minimizing $\mathcal{E}$ ensures that the majority of the embeddings reside within $Z_{s}$.

Noise robustness of the decomposition depends on the orthogonality of $Z_{I}$ and $Z_{T}$ relative to $Z_{s}$. Let $z=z_{s}+z_{I}+z_{T}$ and consider perturbations:

The projections under perturbation satisfy:

The shared subspace $Z_{s}$ forms a geometric locus of alignment, while $Z_{I}$ and $Z_{T}$ act as its orthogonal complements. The effective alignment volume is determined by:

where $\mu_{f}(z)$ and $\mu_{g}(z)$ are the densities of the image and text embeddings projected onto $Z_{s}$.

The shared subspace $Z_{s}$ can be modeled as a low-dimensional manifold within the embedding space $Z$. This manifold captures the semantic “intersection” of image and text modalities, parameterizing cross-modal alignment. Formally, let $Z_{s}$ be a $d_{s}$-dimensional Riemannian manifold embedded in $Z$, such that:

The alignment objective then reduces to finding a mapping $h:Z_{s}\to Z$ that minimizes the alignment error:

This interpretation highlights the geometric constraints imposed by $Z_{s}$, emphasizing the role of manifold regularity in improving alignment performance.

The decomposition $Z=Z_{s}\oplus Z_{I}\oplus Z_{T}$ can also be viewed as a fiber bundle, where $Z_{s}$ serves as the base space and $Z_{I}\times Z_{T}$ as the fiber. Specifically:

Each point in $Z_{s}$ represents a shared semantic embedding, while the fiber $F$ encodes modality-specific deviations. The alignment condition implies that for any $z_{s}\in Z_{s}$:

where $\pi_{s}$ is the projection onto $Z_{s}$.

This interpretation underscores the hierarchical structure of the embedding space, where $Z_{s}$ dictates the global alignment properties and $F$ accommodates modality-specific details.

The shared subspace $Z_{s}$ can also be understood as a fiber variety over a base moduli space. For instance, let $\pi:Z\to M$ be a projection from the embedding space $Z$ to a moduli space $M$, parameterizing semantic categories. Each fiber $\pi^{-1}(m)$ represents embeddings associated with a specific semantic category $m\in M$. The shared subspace $Z_{s}$ then corresponds to the union of fibers aligned across modalities:

This interpretation provides a hierarchical organization of embeddings, where the fiber structure encapsulates modality-specific variations, and the base moduli space captures shared semantic categories.

The geometric interpretations provide guidelines for designing embedding models:

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  A well-regularized $Z_{s}$ improves alignment efficiency, particularly when modeled as a smooth, low-dimensional manifold.

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  Modality-specific subspaces $Z_{I}$ and $Z_{T}$ should be disentangled to avoid interference with the shared semantic space.

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  The fiber bundle structure suggests a hierarchical optimization strategy, first focusing on $Z_{s}$ alignment before refining $Z_{I}$ and $Z_{T}$.

Consider the shared embedding space $Z_{s}$, which can be covered by a collection of open sets $\{U_{\alpha}\}$. A presheaf $\mathcal{F}$ on $Z_{s}$ assigns to each open set $U_{\alpha}$ a set of embeddings $\mathcal{F}(U_{\alpha})$, representing the embeddings consistent with $U_{\alpha}$. For overlapping open sets $U_{\alpha}$ and $U_{\beta}$, the compatibility between embeddings is described by restriction maps:

A presheaf becomes a sheaf if, for any open cover $\{U_{\alpha}\}$ of $U$, the embeddings in $\mathcal{F}(U)$ are uniquely determined by their restrictions to $\mathcal{F}(U_{\alpha})$, satisfying:

The use of the double arrows $\rightrightarrows$ in the sheaf condition highlights the dual projections of local data onto overlapping regions. The first map extracts the restrictions of the local data to the intersections $U_{i}\cap U_{j}$, while the second map applies the same operation but with reversed indexing. The kernel $\ker$ of this pair of maps ensures that local embeddings align consistently across overlaps, enforcing global compatibility within the sheaf framework.

In the context of multimodal alignment, $Z_{s}$ acts as the shared subspace where image and text embeddings are aligned. By modeling $Z_{s}$ with a sheaf $\mathcal{F}$, we ensure the following: Local Consistency: For each open set $U_{\alpha}\subset Z_{s}$, embeddings from different modalities must align locally. Global Compatibility: The local alignments across $Z_{s}$ must be compatible, ensuring that $\mathcal{F}(Z_{s})$ forms a globally consistent shared embedding space.

The shared space $Z_{s}$ can also be viewed as a fiber bundle, where the fibers $\pi^{-1}(m)$ over a moduli point $m\in M$ correspond to embeddings aligned for a specific semantic category. Each fiber represents embeddings with local consistency, while the base moduli space $M$ encodes higher-level semantic categories. Sheaf theory ensures that the embeddings in overlapping fibers $\pi^{-1}(m_{1})$ and $\pi^{-1}(m_{2})$ are globally consistent.