Sigmoid

As explained in Knowledge Graphs and LLMs in Action

A sigmoid activation maps a real-valued score to a value between zero and one, commonly interpreted as a probability for a binary outcome.

A sigmoid activation converts a real-valued score into a probability between zero and one. In binary prediction, this makes a model’s output interpretable as the likelihood of a class or link. section 9.3.1 section 12.2.4

A practical sigmoid-based prediction pipeline follows these steps:

• Generate feature vectors or embeddings for the entities being classified, such as graph nodes or users and movies. section 9.3.1 section 12.2.4
• Compute a score from the representations. Logistic regression uses a linear combination of features, while link prediction uses a user–movie dot product. section 9.3.1 section 12.2.4
• Apply sigmoid to convert the score into a probability, then compare that probability with the binary interaction label during optimization. section 12.2.4

Movie recommendation link prediction

A heterogeneous GNN produces user and movie embeddings. A dot product scores each potential user–movie interaction; sigmoid converts the score into the probability that the link exists. PyTorch’s F.binary_cross_entropy_with_logits combines the sigmoid operation with binary cross-entropy loss for training. section 12.2.4

Figure 12.9 Pipeline for processing heterogeneous graph data in a link prediction system for movie recommendations. The data consists of users and movies as node types and user–movie ratings as edges. Embeddings are generated, with user embeddings learned by the model and movie embeddings initialized from genre features. The data is processed using three heterogeneous GNN encoders (H-GraphConv, H-GAT, and H-SAGE), which aggregate neighbor features to generate node representations. A dot product then quantifies the compatibility between users and movies, with higher scores indicating a greater likelihood of interaction. The scores are finally converted into probabilities representing the likelihood of a link between users and movies.

Follow the complete heterogeneous-graph pipeline from node embeddings and GNN encoders through dot-product compatibility scores and final link probabilities. section 12.2.4

How Knowledge Graphs and LLMs in Action covers it

How deep it goes

Explains score-to-probability conversion and loss integration, but does not derive the sigmoid mathematically.

Examples

Uses graph-node classification and user–movie recommendation link prediction as concrete applications.

What you can do after

Apply it via PyTorch’s F.binary_cross_entropy_with_logits to train probability-producing link-prediction models.

How it connects

Connects logistic regression, dot products, binary cross-entropy loss, graph neural networks, and link prediction.

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Also explained in

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  Grokking Machine LearningA function that maps a real-valued input to a value between zero and one, commonly interpreted as a binary class probability.
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  Inside Deep Learning: Math, Algorithms, ModelsA nonlinear activation function that maps real-valued inputs to a bounded interval.
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  Statistics Every Programmer NeedsA sigmoid function is a function with an S-shaped curve that transitions smoothly between lower and upper limiting values.
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  Grokking StatisticsAn S-curve is a sigmoid-shaped relationship in which growth is initially rapid, then slows toward a plateau.
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  Grokking AI Algorithms, Second EditionA sigmoid function is a bounded S-shaped mathematical function that maps real-valued inputs to values between two limits, commonly zero and one.
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  Data Without LabelsA sigmoid function is a differentiable S-shaped function that maps real-valued inputs to bounded output values and is commonly used to produce binary-classification scores.
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  Natural Language Processing in Action, Second EditionA logistic function is an S-shaped mathematical function that maps an input to a bounded continuous value and provides a soft threshold.
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  Machine Learning for Tabular Data: XGBoost, Deep Learning, and AIA sigmoid is a smooth mathematical function that maps real-valued inputs to values between zero and one.
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  How Large Language Models WorkAn S-curve is a growth pattern in which progress accelerates, reaches a transition region, and then approaches a plateau.
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  Learn Generative AI with PyTorchAn activation function that maps outputs into the range 0 to 1, used here to interpret the discriminator output as a probability.
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  Deep Learning with R, Third EditionA sigmoid activation function maps arbitrary real-valued inputs to values in the interval from zero to one.
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  Machine Learning Algorithms in DepthA sigmoid function is a continuous S-shaped function that maps real-valued inputs to values between zero and one.
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  Deep Learning with Python, Third EditionA statistical classification method that models class probabilities with a linear predictor passed through a logistic function.
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  Math and Architectures of Deep LearningAn S-shaped nonlinear activation function whose steepness and horizontal position can be adjusted with parameters.
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  Reinforcement Learning from Human FeedbackThe sigmoid function maps a real-valued input to a number between zero and one using the function 1/(1+e^-x).

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Related concepts

• activation functionkind of · An activation function maps a neural-network unit’s or layer’s input to an output, often introducing nonlinearity into the network’s computation.
• logistic regressionused by · Logistic regression is a statistical classification model that estimates categorical outcome probabilities by applying a logistic or softmax function to a linear combination of input features.
• language modelused by · A language model is a computational model that represents patterns in linguistic tokens to estimate sequence probabilities and predict or generate language from context.
• probability theoryproduces · Probability theory is the branch of mathematics that formalizes uncertainty and analyzes random events and outcomes using probability measures.
• cross entropy lossused by · Cross-entropy loss measures the expected negative log probability that a predicted distribution assigns to outcomes drawn from a target distribution.

This explainer was generated by Manning's knowledge graph from the text of Knowledge Graphs and LLMs in Action. The section links point to the parts of the book it draws on.