Taylor Decomposition

Description

Taylor Decomposition is a mathematical technique that explains neural network predictions by computing first-order and higher-order derivatives of the network's output with respect to input features. It decomposes the prediction into relevance scores that indicate how much each input feature and feature interaction contributes to the final decision. The method uses Layer-wise Relevance Propagation (LRP) principles to trace prediction contributions backwards through the network layers, providing precise mathematical attributions for each input element.

Example Use Cases

Explainability

Analysing which pixels in an image contribute most to a convolutional neural network's classification decision, showing both positive and negative relevance scores for different regions of the input image.

Understanding how different word embeddings in a sentiment analysis model contribute to the final sentiment score, revealing which terms drive positive vs negative predictions.

Limitations

Mathematically complex requiring deep understanding of calculus and neural network architectures.
Computationally intensive as it requires computing gradients and higher-order derivatives through the entire network.
Approximations used in practice may introduce errors that affect attribution accuracy.
Limited tooling availability compared to other explainability methods, with most implementations being research-focused rather than production-ready.

Resources

Research Papers

Explaining NonLinear Classification Decisions with Deep Taylor Decomposition

Grégoire Montavon et al.•Dec 8, 2015

Nonlinear methods such as Deep Neural Networks (DNNs) are the gold standard for various challenging machine learning problems, e.g., image classification, natural language processing or human action recognition. Although these methods perform impressively well, they have a significant disadvantage, the lack of transparency, limiting the interpretability of the solution and thus the scope of application in practice. Especially DNNs act as black boxes due to their multilayer nonlinear structure. In this paper we introduce a novel methodology for interpreting generic multilayer neural networks by decomposing the network classification decision into contributions of its input elements. Although our focus is on image classification, the method is applicable to a broad set of input data, learning tasks and network architectures. Our method is based on deep Taylor decomposition and efficiently utilizes the structure of the network by backpropagating the explanations from the output to the input layer. We evaluate the proposed method empirically on the MNIST and ILSVRC data sets.

A Rigorous Study Of The Deep Taylor Decomposition

Leon Sixt and Tim Landgraf•Nov 14, 2022

Saliency methods attempt to explain deep neural networks by highlighting the most salient features of a sample. Some widely used methods are based on a theoretical framework called Deep Taylor Decomposition (DTD), which formalizes the recursive application of the Taylor Theorem to the network's layers. However, recent work has found these methods to be independent of the network's deeper layers and appear to respond only to lower-level image structure. Here, we investigate the DTD theory to better understand this perplexing behavior and found that the Deep Taylor Decomposition is equivalent to the basic gradient$\times$input method when the Taylor root points (an important parameter of the algorithm chosen by the user) are locally constant. If the root points are locally input-dependent, then one can justify any explanation. In this case, the theory is under-constrained. In an empirical evaluation, we find that DTD roots do not lie in the same linear regions as the input - contrary to a fundamental assumption of the Taylor theorem. The theoretical foundations of DTD were cited as a source of reliability for the explanations. However, our findings urge caution in making such claims.

Software Packages

lrp_toolbox

May 26, 2016

The LRP Toolbox provides simple and accessible stand-alone implementations of LRP for artificial neural networks supporting Matlab and Python. The Toolbox realizes LRP functionality for the Caffe Deep Learning Framework as an extension of Caffe source code published in 10/2015.

Related Techniques

Name	Description	Assurance Goals
Integrated Gradients	Integrated Gradients is an attribution technique that explains a model's prediction by quantifying the contribution of each input feature. It works by accumulating gradients along a straight path from a user-defined baseline input (e.g., a black image or an all-zero vector) to the actual input. This path integral ensures that the attributions satisfy fundamental axioms like completeness (attributions sum up to the difference between the prediction and the baseline prediction) and sensitivity (non-zero attributions for features that change the prediction). The output is a set of importance scores, often visualised as heatmaps, indicating which parts of the input were most influential for the model's decision.	Explainability
DeepLIFT	DeepLIFT (Deep Learning Important FeaTures) explains neural network predictions by decomposing the difference between the actual output and a reference output back to individual input features. It compares each neuron's activation to a reference activation (typically from a baseline input like all zeros or the dataset mean) and propagates these differences backwards through the network using chain rule modifications. Unlike gradient-based methods, DeepLIFT satisfies the sensitivity property (zero input gets zero attribution) and provides more stable attributions by using discrete differences rather than gradients.	Explainability Transparency
Layer-wise Relevance Propagation	Layer-wise Relevance Propagation (LRP) explains neural network predictions by working backwards through the network to show how much each input feature contributed to the final decision. It follows a simple conservation rule: the total contribution scores always add up to the original prediction. Starting from the output, LRP distributes 'relevance' backwards through each layer using different rules depending on the layer type. This creates a detailed breakdown showing which input features helped or hindered the prediction, making it easier to understand why the network made a particular decision.	Explainability Transparency
Contextual Decomposition	Contextual Decomposition explains LSTM and RNN predictions by decomposing the final hidden state into contributions from individual inputs and their interactions. Unlike simpler attribution methods, it separates the direct contribution of specific words or phrases from the contextual effects of surrounding words. This is particularly useful for understanding how sequential models process language, as it can identify whether a word's influence comes from its individual meaning or from its interaction with nearby words in the sequence.	Explainability Transparency