Supercharging Semantic Search with TS-SS Similarity
As an AI/ML expert, I‘m always on the lookout for powerful yet underutilized algorithms. In this post, I want to share one of my favorite techniques for natural language processing (NLP) tasks like answer retrieval, semantic search, and text matching. It‘s called TS-SS similarity, and it strikes an elegant balance between effectiveness and efficiency.
The Landscape of Text Similarity Metrics
At the heart of many NLP applications is the problem of quantifying the semantic similarity between two snippets of text. Whether you‘re building a chatbot, a search engine, or an automated FAQ system, you need a way to match user queries with the most relevant information in your database.
The classic approach is to represent text numerically using a vector embedding model like word2vec or GloVe. These models learn to map words to points in a high-dimensional space, such that similar words end up close together. By aggregating the word vectors for a piece of text (e.g. by averaging them), we can obtain a single embedding that captures its overall meaning.
With text embeddings in hand, the next step is to define a similarity function between them. The most common choices are cosine similarity and Euclidean distance.
Cosine similarity measures the angle between two vectors, ignoring their magnitudes:
$\cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|}$
where $\mathbf{A}$ and $\mathbf{B}$ are the embeddings and $|\cdot|$ denotes the L2 norm.
Euclidean distance, on the other hand, is the straight-line distance between two points in space:
$d(\mathbf{A},\mathbf{B}) = \sqrt{\sum_{i=1}^n (A_i – B_i)^2}$
Where $n$ is the dimensionality of the embeddings.
In practice, cosine similarity is more commonly used due to its nice properties like boundedness in $[-1, 1]$ and insensitivity to vector lengths. However, it also has some well-known drawbacks:
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Cosine similarity can assign high scores to vectors that point in the same direction but have very different magnitudes. This is problematic for NLP because text length often correlates with information content.
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The angular distance between vectors is not always perceptually meaningful. For example, the angles between the words "cat", "dog", and "basketball" are roughly equal, even though the first two are much more semantically similar.
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Cosine similarity is not a proper metric since it violates the triangle inequality. This can make it tricky to use in downstream algorithms that rely on metric space properties.
Euclidean distance sidesteps some of these issues but has its own limitations. In particular, it‘s very sensitive to the absolute positions of vectors rather than just their relative angles. This means that small offsets or rotations in the embedding space can lead to large changes in distance.
Introducing TS-SS Similarity
To address the shortcomings of cosine and Euclidean similarity, a team at the University of Tehran introduced a new metric called TS-SS: Triangle Area and Sector Area Similarity [1]. The key idea is to combine two different geometric notions of similarity in vector space.
First, let‘s visualize two text embeddings as vectors $\mathbf{A}$ and $\mathbf{B}$ originating from the same point:
The Triangle Area Similarity (TS) looks at the area of the triangle formed by connecting the heads of $\mathbf{A}$ and $\mathbf{B}$. Intuitively, the smaller this area, the more similar the vectors are in terms of direction. Mathematically, TS is defined as:
$\operatorname{TS}(\mathbf{A},\mathbf{B}) = \frac{1}{2} |\mathbf{A}| |\mathbf{B}| \sin(\theta‘)$
where $\theta‘ = \arccos\left(\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|}\right) + \frac{\pi}{18}$. The extra $10^\circ$ in $\theta‘$ ensures that TS is non-zero even for very similar vectors.
Next, the Sector Area Similarity (SS) considers both the direction and magnitude differences between $\mathbf{A}$ and $\mathbf{B}$. It first calculates the Euclidean distance $d$ and magnitude difference $m$:
$d(\mathbf{A},\mathbf{B}) = \sqrt{\sum_{i=1}^n (A_i – B_i)^2}$
$m(\mathbf{A},\mathbf{B}) = \big| |\mathbf{A}| – |\mathbf{B}| \big|$
Then, it defines the sector area in terms of $d$, $m$, and $\theta‘$:
$\operatorname{SS}(\mathbf{A},\mathbf{B}) = \frac{\theta‘}{2} (d+m)^2$
Here‘s a visual depiction of the sector area:
Putting it all together, the TS-SS similarity is simply the product of TS and SS:
$\operatorname{TS-SS}(\mathbf{A},\mathbf{B}) = \operatorname{TS}(\mathbf{A},\mathbf{B}) \cdot \operatorname{SS}(\mathbf{A},\mathbf{B})$
By combining these two complementary notions of similarity, TS-SS provides a robustness to variations in both direction and magnitude. It satisfies all the metric space axioms while being bounded in $[0, 1]$.
Implementing TS-SS in Python
Thanks to NumPy and SciPy, implementing TS-SS in Python is straightforward. Here‘s a minimal working example:
import numpy as np
from scipy.spatial.distance import euclidean
def ts_ss(A, B):
cosine = np.dot(A,B) / (np.linalg.norm(A) * np.linalg.norm(B))
theta_prime = np.arccos(cosine) + np.radians(10)
triangle_area = 0.5 * np.linalg.norm(A) * np.linalg.norm(B) * np.sin(theta_prime)
magnitude_diff = np.abs(np.linalg.norm(A) - np.linalg.norm(B))
euclidean_dist = euclidean(A, B)
sector_area = (theta_prime / 2) * (euclidean_dist + magnitude_diff)**2
return triangle_area * sector_areaTo use this in an NLP pipeline, you‘d first convert your text inputs to vector embeddings (e.g. using spaCy), then pass the embeddings to ts_ss. For example, to find the most similar answer to a user query:
import spacy
nlp = spacy.load("en_core_web_md")
def text2vec(text):
return nlp(text).vector
query = "What is the capital of France?"
answers = [
"Paris is the capital of France.",
"France‘s capital is called Paris.",
"Berlin is the capital of Germany.",
]
query_vec = text2vec(query)
answer_vecs = [text2vec(a) for a in answers]
similarities = [ts_ss(query_vec, a) for a in answer_vecs]
best_answer_index = np.argmin(similarities)
print(answers[best_answer_index])This will output "Paris is the capital of France." as the best match to the query.
Of course, there are many ways to optimize this code for efficiency and scalability. Some ideas:
- Precompute the embeddings for your answer corpus and store them on disk or in a database
- Use an Approximate Nearest Neighbors (ANN) library like Annoy or FAISS to quickly find the most similar answers
- Parallelize the similarity calculations across multiple CPU cores or GPUs
I encourage you to experiment with TS-SS in your own projects and see how it compares to other similarity metrics. In my experience, it often outperforms cosine similarity while being much faster than more complex models like BERT.
Evaluating TS-SS on Benchmark Datasets
To rigorously test the effectiveness of TS-SS, the original authors conducted experiments on three standard text classification datasets [1]:
- 20Newsgroups: 18,828 newsgroup posts on 20 topics
- Reuters-21578: 10,788 news articles grouped into 8 categories
- WebKB: 4,518 university webpages labeled with 7 categories
For each dataset, they compared the accuracy of TS-SS to cosine similarity and Euclidean distance in a k-Nearest Neighbors (kNN) classification setup. The results are summarized below:
| Method | 20Newsgroups | Reuters | WebKB |
|---|---|---|---|
| TS-SS | 96.1% | 97.2% | 91.3% |
| Cosine Similarity | 84.3% | 93.1% | 71.4% |
| Euclidean Distance | 68.7% | 89.6% | 56.5% |
As we can see, TS-SS consistently outperforms the baselines, often by a significant margin. The gains are particularly pronounced on the WebKB dataset, which has a small number of training examples per class. This suggests that TS-SS is able to capture meaningful semantic relationships even in data-scarce scenarios.
Digging deeper, the authors also visualized the learned embeddings using t-SNE, a technique for projecting high-dimensional vectors into 2D space. Here are the plots for the 20Newsgroups dataset:
The TS-SS embeddings show much cleaner separation between the newsgroup topics, while cosine similarity and Euclidean distance have a lot more overlap and scatter. This makes intuitive sense given that TS-SS is better able to balance document length and topic specificity.
Future Directions and Applications
Despite its strong empirical performance, there‘s still much to be explored with TS-SS and semantic similarity more broadly. One interesting direction is learning the metric end-to-end rather than relying on fixed embedding functions. Recent work has shown promising results for supervised similarity learning using techniques like siamese networks and triplet loss [2].
Another exciting area is extending TS-SS to non-Euclidean geometries like hyperbolic space. Hyperbolic embeddings have been shown to better capture hierarchical and asymmetric relationships [3], which are common in natural language. A hyberbolic version of TS-SS could potentially combine the best of both worlds.
Beyond the core algorithm, there are also numerous practical applications of semantic similarity that could benefit from TS-SS. For example:
- Plagiarism Detection: Efficiently comparing new documents to a large corpus of existing work
- Semantic Search: Retrieving relevant documents based on conceptual rather than exact keyword matches
- Question Answering: Finding the most relevant passages in a knowledge base to answer open-ended queries
- Recommendation Systems: Discovering related items based on user interaction and review text
- Dialogue Systems: Selecting appropriate responses based on user input and conversation context
Conclusion
As an AI researcher and practitioner, I believe TS-SS is a valuable addition to the NLP toolkit. By combining the strengths of cosine similarity and Euclidean distance in a principled way, it offers a robust and efficient solution to a core problem in natural language understanding.
Of course, TS-SS is not a silver bullet – the choice of similarity metric always depends on the task at hand. But for many common scenarios like semantic search and text classification, TS-SS is definitely worth considering. I hope this deep dive has piqued your curiosity to learn more!
If you‘re interested in implementing TS-SS in your own projects, I recommend starting with the sample code in this post and experimenting on a domain-specific dataset. Feel free to reach out if you have any questions or insights to share.
