Academic base of the mathematical block

Content and key concepts

Date of receipt and inspection: 2026-09-09. The study was completed before the pages were implemented. The training examples are self-examples; the receipt of the source does not imply the replication of its experiments. The numbering of the sources M01-M09 refers to the whole block so as not to reproduce the same bibliographic records.
Issues and map of evidence
Chapter
Research issues
Support
01
What's space? When a set is a base? What standards are correct? How dot is connected to the angle? When cosine is not defined? Why does projection minimize the rest? What changes when units change?
M01, M02; projection through orthogonal balance
02
What do the columns of the matrix define? How do you get row-colourn product? Why AB isn't BA? What measures rank? When inverse doesn't exist? Is there always real eigenvectors? How does the SVD differ from eigendecomposition? What low-rank saves and what does not guarantee?
M01, M02, M08
03
What's the derivative different from the skeleton? What does the partial see? Why does the gradiant maximize? What form Jacobian? Why do you need to transfer to backward? What says Hessian and when the test doesn't solve the question? How does chain rule work? How is fiinite diffusion different from autograd?
M03, M05; own analytical findings
04
What's normal in PMF? How does density differ from probability? How do you get Bayes out? Why isn't it posterior? Where is the logarithm of entropy? When the CE is infinite? Why does KL not distance? How can you define perplexity without confusion?
M04, M07; definition of PP and algebra geometric mean
05
Why does the anti-gradient locally reduce the loss? How do you get the step limit for the square? How does SGD differ from the GD? How is the batch? What exactly does the optimizer keep? Why bias control? Where is the epsilon? What is the clipping different from the Decay?
M05, M06, M09; Conclusions from selected formulas
Sources
M01. Deep Learning, Chapter 2
Ian Goodfellow, Yoshua Bengio, Aaron Courville.
Deep Learning
, MIT Press, 2016 University textbook.
https://www.deeplearningbook.org/contents/linear_algebra.html
The substantive text of the chapter has been obtained: definitions, works, span/independence, rules (§2.5), eigendecomposition (§2.7), SVD (§2.8), determinant and projection output in PCA (§2.12). HTML conversions damage the signs, so our formulas are written independently. The full spectral theorem machine is not re-proved. Singular directions may include reflections, not just bends. Symmetric matrices are used for positive certainty.
M02. Dive into Deep Learning, Linear Algebra
Aston Zhang, Zachary C. Lipton, Mu Li, Alexander J. Smola. Cambridge University Press, 2023, web 1.0.3.
URL:
https://d2l.ai/_sources/chapter_preliminaries/linear-algebra.rst.txt
Received by RST; the output of the tool is cut, the visible part confirms scalars, vectors, shape, transport and the distinction of Hadamard/matmul. Late sections that do not appear in the visible answer are not considered to have been read. Matmul and the standards are further checked on M01. The code with the frames has not been started.
M03. Dive into Deep Learning, Calculus
The same authors, 2023, §2.4.
https://d2l.ai/_sources/chapter_preliminaries/calculus.rst.txt
The definition of derivative, limits, partial derivatives, gradient and chain root with several ways is also used in the source, using a transposed layout derivative vector function; here, the jacobian J_ij is explicitly accepted =f_i/x_j, shape [exit, exit]. Therefore, the backward is recorded by J\T g, rather than copying the formula without the Shapes test.
M04. Deep Learning, Probability and Information Theory
Goodfellow, Bengio, Courville, 2016, Chapter 3.
https://www.deeplearningbook.org/contents/prob.html
Text obtained; visible §3.3 - 3.8, §3.11 Bayes, § 3.13 entropy/KL/CE. The end of the conclusion is cut. This is a secondary academic source, not the original Shannon. The distinct division of the name and PMF, correlation and independence, likelihood and distribution by parameter. The units here are nats; continous disferential entrepy is not replaced by discrete.
M05. Dive into Deep Learning, Gradient Descent
Zhang et al., 2023, §12.3.
URL:
https://d2l.ai/_sources/chapter_optimization/gd.rst.txt
Got RST with Taylor motivation, lightning rate, multidimensional GD, Newton/Hessian. We don't tolerate a wrong source terminology that calls a square convergence "line": in the next step, the error square is called quadratic estimate. We do not claim that a conventional binary search solves any liine search. The own examples use positive diagonal square.
M06. Dive into Deep Learning, Adam
Zhang et al., 2023, §12.10.
URL:
https://d2l.ai/_sources/chapter_optimization/adam.rst.txt
The equations EMA, bias control, update and examples of implementation have been obtained.
Discrepancies:
The formula and PyTorch-example have epsilon inside the denominator after sqrt; in the shown TensorFlow-fragment
+eps
The second point is not called dispersion without subtracting the average square. In our epsilon labs = 1e-8 outside sqrt, but within the denominator; this is a clear choice, not a statement of the identity of all libraries.
M07. Dive into Deep Learning, Information Theory
Chang et al., 2023, mathematical application.
https://d2l.ai/_sources/chapter_appendix-mathematics-for-deep-learning/information-theory.rst.txt
The definitions of enropy, KL and CE=H+KL have been obtained. The conclusion has been cut in late.
We don't copy the wrong demonstrations:
The arrays of normal random numbers with negative values are not PMF, but
abs(KL)
The source uses bits, default nats; the translation factor is specified.
M08. LoRA: Low-Rank Adaptation of Large Language Models
Edward J. Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, Weizhu Chen, 2021.
URL:
https://arxiv.org/abs/2106.09685
Primary preprint; received
abstact and metadata only v2
. Supports the freezing of pre-traded lights and the training of Frank-decomposition matrices. Authors ' quantitative gains are not promised for an arbitrary network. The BA formula and its own parameter score are based on shapes; the complete training recipe LoRA is not declared by the researched.
M09. Adam: A Method for Stochastic Optimization
Diederik P. Kingma, Jimmy Ba, ICLR 2015; preprint 2014, card v9 (2017).
https://arxiv.org/abs/1412.6980
Primary work; received
abstract/metadata only
. Supports the origin of the algorithm and the use of moment estimates. The formulas are checked by M06 content and independent oracle, not one abstact. Adam universal convergence is not approved. M04/M07 is used for probabilistic definitions of chapter 04, not this paper.
Limitations
This is a targeted overview of the basics of two independent textbooks and two primary abstracts, rather than eight independent works per chapter. The sections of one book are not considered independent authors. For basic mathematical titles, additio