Publication
] A Perspective: Sparse Compositionality and Efficiently Computable Intelligence. (2026).
Perspective_SPCOMP-9.pdf (170.23 KB)
Perspective_SPCOMP-9.pdf (170.23 KB) 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)
v1.0 (984.15 KB) v5.7 adding in context learning etc (1.16 MB) Explicit regularization and implicit bias in deep network classifiers trained with the square loss. arXiv (2020). at <https://arxiv.org/abs/2101.00072>
Compositional sparsity of learnable functions. Bulletin of the American Mathematical Society 61, 438-456 (2024).
From Marr’s Vision to the Problem of Human Intelligence. (2021). CBMM-Memo-118.pdf (362.19 KB)
CBMM-Memo-118.pdf (362.19 KB) Turing++ Questions: A Test for the Science of (Human) Intelligence. AI Magazine 37 , 73-77 (2016). Turing_Plus_Questions.pdf (424.91 KB)
Turing_Plus_Questions.pdf (424.91 KB) On efficiently computable functions, deep networks and sparse compositionality. (2025). Deep_sparse_networks_approximate_efficiently_computable_functions.pdf (223.15 KB)
Deep_sparse_networks_approximate_efficiently_computable_functions.pdf (223.15 KB) Loss landscape: SGD has a better view. (2020). CBMM-Memo-107.pdf (1.03 MB) Typos and small edits, ver11 (955.08 KB) Small edits, corrected Hessian for spurious case (337.19 KB)
CBMM-Memo-107.pdf (1.03 MB) Typos and small edits, ver11 (955.08 KB) Small edits, corrected Hessian for spurious case (337.19 KB) 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)
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) What if.. (2015). What if.pdf (2.09 MB)
What if.pdf (2.09 MB) Associative Memory as the Core of Intelligence in Technology and Evolution. (2026). Review_On_Associative_Memories-14.pdf (245.78 KB)
Review_On_Associative_Memories-14.pdf (245.78 KB) Deep Learning: mathematics and neuroscience. (2016). Deep Learning- mathematics and neuroscience.pdf (1.25 MB)
Deep Learning- mathematics and neuroscience.pdf (1.25 MB) Double descent in the condition number. (2019). Fixing typos, clarifying error in y, best approach is crossvalidation (837.18 KB) Incorporated footnote in text plus other edits (854.05 KB) Deleted previous discussion on kernel regression and deep nets: it will appear, extended, in a separate paper (795.28 KB) correcting a bad typo (261.24 KB) Deleted plot of condition number of kernel matrix: we cannot get a double descent curve (769.32 KB)
Fixing typos, clarifying error in y, best approach is crossvalidation (837.18 KB) Incorporated footnote in text plus other edits (854.05 KB) Deleted previous discussion on kernel regression and deep nets: it will appear, extended, in a separate paper (795.28 KB) correcting a bad typo (261.24 KB) Deleted plot of condition number of kernel matrix: we cannot get a double descent curve (769.32 KB) Visual Cortex and Deep Networks: Learning Invariant Representations. 136 (The MIT Press, 2016). at <https://mitpress.mit.edu/books/visual-cortex-and-deep-networks>
Implicit dynamic regularization in deep networks. (2020). v1.2 (2.29 MB) v.59 Update on rank (2.43 MB)
v1.2 (2.29 MB) v.59 Update on rank (2.43 MB) 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)
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) Deep Leaning: Mathematics and Neuroscience. A Sponsored Supplement to Science Brain-Inspired intelligent robotics: The intersection of robotics and neuroscience, 9-12 (2016).
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)
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) I-theory on depth vs width: hierarchical function composition. (2015). cbmm_memo_041.pdf (1.18 MB)
cbmm_memo_041.pdf (1.18 MB) Complexity Control by Gradient Descent in Deep Networks. Nature Communications 11, (2020). s41467-020-14663-9.pdf (431.68 KB)
s41467-020-14663-9.pdf (431.68 KB) Compositional Sparsity of Learnable Functions. (2024). This is an update of the AMS paper (230.72 KB)
This is an update of the AMS paper (230.72 KB) Theory II: Deep learning and optimization. Bulletin of the Polish Academy of Sciences: Technical Sciences 66, (2018). 03_775-788_00920_Bpast.No_.66-6_31.12.18_K2.pdf (5.43 MB)
03_775-788_00920_Bpast.No_.66-6_31.12.18_K2.pdf (5.43 MB) Cervelli menti algoritmi. 272 (Sperling & Kupfer, 2023). at <https://www.sperling.it/libri/cervelli-menti-algoritmi-marco-magrini>
Computational role of eccentricity dependent cortical magnification. (2014). CBMM-Memo-017.pdf (1.04 MB)
CBMM-Memo-017.pdf (1.04 MB) From Associative Memories to Powerful Machines. (2021). v1.0 (1.01 MB) v1.3Section added August 6 on self attention (3.9 MB)
v1.0 (1.01 MB) v1.3Section added August 6 on self attention (3.9 MB)