# Stable Recovery of Entangled Weights: Towards Robust Identification of Deep Neural Networks from Minimal Samples

arXiv: https://arxiv.org/abs/2101.07150 · HTML: https://mlsanity.com/paper/2101.07150/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 96 references that are in the map.
- First posted on arXiv 18 Jan 2021.

## Builds on (most cited first, 30 of 51)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [The State of Sparsity in Deep Neural Networks](https://mlsanity.com/paper/1902.09574/) (2019)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Deep Speech: Scaling up end-to-end speech recognition](https://mlsanity.com/paper/1412.5567/) (2014)
- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Towards moderate overparameterization: global convergence guarantees for training shallow neural networks](https://mlsanity.com/paper/1902.04674/) (2019)
- [Convergence Analysis of Two-layer Neural Networks with ReLU Activation](https://mlsanity.com/paper/1705.09886/) (2017)
- [Theoretical insights into the optimization landscape of over-parameterized shallow neural networks](https://mlsanity.com/paper/1707.04926/) (2017)
- [Recovery Guarantees for One-hidden-layer Neural Networks](https://mlsanity.com/paper/1706.03175/) (2017)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Optimal Approximation with Sparsely Connected Deep Neural Networks](https://mlsanity.com/paper/1705.01714/) (2017)
- [Reverse-Engineering Deep ReLU Networks](https://mlsanity.com/paper/1910.00744/) (2019)
- [Small ReLU networks are powerful memorizers: a tight analysis of memorization capacity](https://mlsanity.com/paper/1810.07770/) (2018)
- [Implicit Bias in Deep Linear Classification: Initialization Scale vs Training Accuracy](https://mlsanity.com/paper/2007.06738/) (2020)
- [Optimization for deep learning: theory and algorithms](https://mlsanity.com/paper/1912.08957/) (2019)
- [Good Subnetworks Provably Exist: Pruning via Greedy Forward Selection](https://mlsanity.com/paper/2003.01794/) (2020)
- [Global Convergence of Deep Networks with One Wide Layer Followed by Pyramidal Topology](https://mlsanity.com/paper/2002.07867/) (2020)
- [Deep vs. shallow networks : An approximation theory perspective](https://mlsanity.com/paper/1608.03287/) (2016)

## Built on by (most cited first, 8 of 8)

- [Sampling weights of deep neural networks](https://mlsanity.com/paper/2306.16830/) (2023)
- [Identifiability of Deep Polynomial Neural Networks](https://mlsanity.com/paper/2506.17093/) (2025)
- [Landscape analysis of an improved power method for tensor decomposition](https://mlsanity.com/paper/2110.15821/) (2021)
- [Learning ReLU networks to high uniform accuracy is intractable](https://mlsanity.com/paper/2205.13531/) (2022)
- [Finite Sample Identification of Wide Shallow Neural Networks with Biases](https://mlsanity.com/paper/2211.04589/) (2022)
- [Sequencing the Neurome: Towards Scalable Exact Parameter Reconstruction of Black-Box Neural Networks](https://mlsanity.com/paper/2409.19138/) (2024)
- [Finite Samples for Shallow Neural Networks](https://mlsanity.com/paper/2503.12744/) (2025)
- [Curvature Cryptanalysis of Smooth Transformer Feed-Forward Networks](https://mlsanity.com/paper/2608.28843/) (2026)

## Abstract (arXiv)

In this paper we approach the problem of unique and stable identifiability of generic deep artificial neural networks with pyramidal shape and smooth activation functions from a finite number of input-output samples. More specifically we introduce the so-called entangled weights, which compose weights of successive layers intertwined with suitable diagonal and invertible matrices depending on the activation functions and their shifts. We prove that entangled weights are completely and stably approximated by an efficient and robust algorithm as soon as mathcal O(D2 × m) nonadaptive input-output samples of the network are collected, where D is the input dimension and m is the number of neurons of the network. Moreover, we empirically observe that the approach applies to networks with up to mathcal O(D × mL) neurons, where mL is the number of output neurons at layer L. Provided knowledge of layer assignments of entangled weights and of remaining scaling and shift parameters, which may be further heuristically obtained by least squares, the entangled weights identify the network completely and uniquely. To highlight the relevance of the theoretical result of stable recovery of entangled weights, we present numerical experiments, which demonstrate that multilayered networks with generic weights can be robustly identified and therefore uniformly approximated by the presented algorithmic pipeline. In contrast backpropagation cannot generalize stably very well in this setting, being always limited by relatively large uniform error. In terms of practical impact, our study shows that we can relate input-output information uniquely and stably to network parameters, providing a form of explainability. Moreover, our method paves the way for compression of overparametrized networks and for the training of minimal complexity networks.