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Poggio, T. & Fraser, M. Compositional sparsity of learnable functions. Bulletin of the American Mathematical Society 61, 438-456 (2024).

Poggio, T. A. & Xu, M. On efficiently computable functions, deep networks and sparse compositionality. (2025).

Deep_sparse_networks_approximate_efficiently_computable_functions.pdf (223.15 KB)

Poggio, T., Liao, Q. & Xu, M. Implicit dynamic regularization in deep networks. (2020).

v1.2 (2.29 MB)

v.59 Update on rank (2.43 MB)

Poggio, T. Stable Foundations for Learning: a framework for learning theory (in both the classical and modern regime). (2020).

Original file (584.54 KB)

Corrected typos and details of "equivalence" CV stability and expected error for interpolating machines. Added Appendix on SGD. (905.29 KB)

Edited Appendix on SGD. (909.19 KB)

Deleted Appendix. Corrected typos etc (880.27 KB)

Added result about square loss and min norm (898.03 KB)

Poggio, T. & Meyers, E. Turing++ Questions: A Test for the Science of (Human) Intelligence. AI Magazine 37 , 73-77 (2016).

Turing_Plus_Questions.pdf (424.91 KB)

Poggio, T., Mhaskar, H., Rosasco, L., Miranda, B. & Liao, Q. Theory I: Why and When Can Deep Networks Avoid the Curse of Dimensionality?. (2016).

CBMM-Memo-058v1.pdf (2.42 MB)

CBMM-Memo-058v5.pdf (2.45 MB)

CBMM-Memo-058-v6.pdf (2.74 MB)

Proposition 4 has been deleted (2.75 MB)

Poggio, T., Liao, Q. & Banburski, A. Complexity Control by Gradient Descent in Deep Networks. Nature Communications 11, (2020).

s41467-020-14663-9.pdf (431.68 KB)

Poggio, T. & Liao, Q. Theory I: Deep networks and the curse of dimensionality. Bulletin of the Polish Academy of Sciences: Technical Sciences 66, (2018).

02_761-774_00966_Bpast.No_.66-6_28.12.18_K1.pdf (1.18 MB)

Poggio, T. What if.. (2015).

What if.pdf (2.09 MB)

Poggio, T. Deep Learning: mathematics and neuroscience. (2016).

Deep Learning- mathematics and neuroscience.pdf (1.25 MB)

Poggio, T. From Associative Memories to Powerful Machines. (2021).

v1.0 (1.01 MB)

v1.3Section added August 6 on self attention (3.9 MB)

Poggio, T. & Anselmi, F. Visual Cortex and Deep Networks: Learning Invariant Representations. 136 (The MIT Press, 2016). at <https://mitpress.mit.edu/books/visual-cortex-and-deep-networks>

Poggio, T. & Liao, Q. Explicit regularization and implicit bias in deep network classifiers trained with the square loss. arXiv (2020). at <https://arxiv.org/abs/2101.00072>

Poggio, T. Deep Leaning: Mathematics and Neuroscience. A Sponsored Supplement to Science Brain-Inspired intelligent robotics: The intersection of robotics and neuroscience, 9-12 (2016).

Poggio, T. How Deep Sparse Networks Avoid the Curse of Dimensionality: Efficiently Computable Functions are Compositionally Sparse. (2022).

v1.0 (984.15 KB)

v5.7 adding in context learning etc (1.16 MB)

Poggio, T., Anselmi, F. & Rosasco, L. I-theory on depth vs width: hierarchical function composition. (2015).

cbmm_memo_041.pdf (1.18 MB)

Poggio, T. From Marr’s Vision to the Problem of Human Intelligence. (2021).

CBMM-Memo-118.pdf (362.19 KB)

Poggio, T. & Banburski, A. An Overview of Some Issues in the Theory of Deep Networks. IEEJ Transactions on Electrical and Electronic Engineering 15, 1560 - 1571 (2020).

Poggio, T. & Squire, L. R. The History of Neuroscience in Autobiography Volume 8 8, (Society for Neuroscience, 2014).

Volume Introduction and Preface (232.8 KB)

TomasoPoggio.pdf (1.43 MB)

Poggio, T. & Fraser, M. Compositional Sparsity of Learnable Functions. (2024).

This is an update of the AMS paper (230.72 KB)

Poggio, T., Banburski, A. & Liao, Q. Theoretical issues in deep networks. Proceedings of the National Academy of Sciences 201907369 (2020). doi:10.1073/pnas.1907369117

PNASlast.pdf (915.3 KB)

Poggio, T., Banburski, A. & Liao, Q. Theoretical Issues in Deep Networks. (2019).

CBMM Memo 100 v1 (1.71 MB)

CBMM Memo 100 v3 (8/25/2019) (1.31 MB)

CBMM Memo 100 v4 (11/19/2019) (1008.23 KB)

Poggio, T. et al. Theory of Deep Learning III: explaining the non-overfitting puzzle. (2017).

CBMM-Memo-073.pdf (2.65 MB)

CBMM Memo 073 v2 (revised 1/15/2018) (2.81 MB)

CBMM Memo 073 v3 (revised 1/30/2018) (2.72 MB)

CBMM Memo 073 v4 (revised 12/30/2018) (575.72 KB)

Poggio, T. & Magrini, M. Cervelli menti algoritmi. 272 (Sperling & Kupfer, 2023). at <https://www.sperling.it/libri/cervelli-menti-algoritmi-marco-magrini>

Poggio, T. & Cooper, Y. Loss landscape: SGD has a better view. (2020).