# Alignment Score

Given a data point $$\mathbf{x} \in \mathcal{X}$$ uploaded in response to instruction $$\mathbf{y} \in \mathcal{Y}$$, the alignment score $$s(\mathbf{x}, \mathbf{y})$$ quantifies how well the response data point adheres to the requirements set by the instruction.

$$
s(\mathbf{x}, \mathbf{y}) : \mathcal{X} \times \mathcal{Y} \to \[0, 1]
$$

For ease of exposition, we will first assume that the instructions are free-form text, while the data uploaded in response is one of image, audio, video, or text. Later, we also show how these building blocks are combined to support multi-modal instructions and data.

At a high level, the idea is to map both the user post $$\mathbf{x}$$ and the instruction $$\mathbf{y}$$ to a joint embedding space that fosters the proximity of semantically similar samples and separation of dissimilar ones. Then we can simply compute the alignment score using a suitable distance measure $$\mathcal{D}$$, defined over the joint embedding space.

Specifically, we assume access to mapping functions (encoders): $$f\_w^{\mathcal{X}}(\cdot): \mathcal{X} \to \mathcal{H}$$ and $$f\_w^{\mathcal{Y}}(\cdot): \mathcal{Y} \to \mathcal{H}$$, mapping data and instruction to a joint embedding space where $$\mathcal{H} \in \mathbb{R}^k$$ denotes a separable Hilbert Space equipped with inner product operation.

Given an instruction $$\mathbf{y} \in \mathcal{Y}$$ and corresponding user-uploaded data $$\mathbf{x} \in \mathcal{X}$$, we can compute the embedding vectors (representation in the joint embedding space):

$$
\mathbf{z}*{\mathbf{x}} = f\_w^{\mathcal{X}}(\mathbf{x}), \quad \mathbf{z}*{\mathbf{y}} = f\_w^{\mathcal{Y}}(\mathbf{y})
$$

Once we have vector representations of the instruction and corresponding sample on the shared embedding space, we measure alignment as the cosine similarity (normalized inner product) with a Rectified Linear Unit (ReLU) activation.

$$
s(\mathbf{x}, \mathbf{y}) = \max \left(0, \frac{1}{\tau} \frac{\mathbf{z}*{\mathbf{x}}^T \mathbf{z}*{\mathbf{y}}}{|\mathbf{z}*{\mathbf{x}}| |\mathbf{z}*{\mathbf{y}}|}\right)
$$

Intuitively, we project the sample and instruction onto a hypersphere $$\mathcal{S}^{k-1}\_1 = {\mathbf{z} \in \mathbb{R}^k : |\mathbf{z}|=\frac{1}{\tau}}$$ and compute the *angular distance* between the semantic embeddings of the sample and instruction. Here, $$\tau \in \mathbb{R}^+$$ is a hyper-parameter that balances the spread of the representations on $$\mathcal{S}^{k-1}\_1$$. 


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