graph LR idea_06851353_d0ff_4de8_981d_022050eac719["Orthogonal Eigenvectors for Distinct Eigenvalues in Symmetric Matrices"] idea_0dbae8aa_af97_4e8f_8c22_ea074673ad0b["Matrix Multiplication Algorithms and Properties in Linear Algebra"] idea_0dccfd4a_bb1d_4d4d_9d7e_36cfd1509b87["Jacobian Matrix of Partial Derivatives in Multivariable Calculus"] idea_105750aa_214b_4913_9b4e_dfd41fd2ed85["The Gradient Vector Collects the First Partial Derivatives"] idea_11af8705_6a18_4a98_9b64_02b3145f956c["Learning-Rate Warmup as a Growing Step Size in Transformer Training"] idea_11c10d16_59a8_46c3_96be_9925f4fa30e6["Backpropagation"] idea_152a5426_e186_4377_bbab_6addd7f7966f["Outer Product of Two Vectors in Linear Algebra"] idea_169779a8_f66e_4265_88af_918ed56564fe["Low-Rank Factorization Caps the Rank of QK^T at d_k in Attention"] idea_16dd60da_c5de_4e82_9413_78dff90cfa85["Oversmoothing and Rank Collapse as the Steady State of Attention Diffusion"] idea_18bd63d3_d2a3_45f1_a05f_367c4625ead4["Gradient Clipping"] idea_1ab16196_1504_4e2e_a184_afa8f3ed0a25["Scaled Dot-Product Attention as QK^T over the Square Root of d"] idea_1b4c5424_3f70_45ab_b2eb_fb7af1963c4d["Cross-Entropy Gradient With Respect to Logits Equals p Minus y"] idea_1d3fccba_cec3_4106_96ef_240f17d77aa9["Multivariable Chain Rule for Composite Functions"] idea_1ebe495d_9a13_412d_8459_a82f23b0fadf["Derivatives"] idea_20feb64e_1a7d_4901_872e_c8516a8b359e["Residual Connections Keep Each Block's Jacobian Near the Identity"] idea_2132fb22_760b_47ad_a3f5_b8bea6818535["Matrix Multiplication Rules in Linear Algebra"] idea_21e80959_f5e3_4c74_9f10_bedc5b9f3504["Tokens as Interacting Particles Whose Density Obeys a Continuity Equation"] idea_250be30b_2ba3_4625_9c5c_3697ece9f37b["Fokker-Planck Equations in Stochastic Dynamics"] idea_28f8e056_82d8_4577_856d_73fb3875a368["Rotation Matrices"] idea_2dcf2705_32c5_4dc5_9965_3cca1bb09b63["Matrix Product Gradients A^TG and GB^T in Backpropagation"] idea_2e814f08_a86d_4308_acc1_f0419bdb9da0["Matrix Rank Definition in Linear Algebra"] idea_2e8edc82_1116_413d_a0a6_35e55543f553["Variance"] idea_2fa29f8e_6b5f_44ba_8133_ade5fa6f4cf0["Derivative"] idea_3b5c9594_bb5c_4337_95b5_aabf0cdac27f["Scalar Multiplication and Matrix Scaling in Linear Algebra"] idea_3c6874e9_d0f7_442c_b1b8_08cf55e56203["Softmax Over Logits as a Next-Token Distribution"] idea_3d330681_e2dd_4e32_b554_a00eb123431b["Deriving the Normal Distribution's PDF, Mean, and Variance in Probability"] idea_3db74e87_2793_4bad_bc9a_24bdef5ee432["Maximum Likelihood Estimation"] idea_3e805dc8_1e55_4156_ac55_f363c0ff3cb1["Vanishing Gradient Problem"] idea_40c308c7_8691_447d_96d3_ade271a0991c["Representing a Computation as a Directed Computational Graph"] idea_425c3778_1a9c_4262_9b1f_0e41fb1a33a6["Linear Independence Verification in Vector Sets"] idea_42b61ac3_a40f_42bb_99ff_00bcf094e419["Geometric Linear Transformations in R2: Reflection, Rotation, Shear, and Projection Matrices"] idea_4cd5b655_dc4d_4bac_8efd_a9d8eacee3a7["Feedforward Networks (MLPs) in Transformers"] idea_54e415ff_3c0b_4e5d_b237_b0015316d862["Learning-Rate Stability Limit Set by the Largest Hessian Eigenvalue"] idea_55daa1fa_5d15_419b_af55_c3bf09a43af9["Gradient Flows and Gradient-Like Dynamical Systems"] idea_58ef03a7_696d_4ace_a108_26f6241d2864["Gradient Descent"] idea_5d206c58_1c31_4b26_bef4_5a61d8b028f8["Loss Functions"] idea_5f68aa37_e831_49c9_a5ec_b8143369eca1["Rotary Position Embeddings (RoPE)"] idea_5f964e9f_1680_4dd2_9522_7ffe33552992["Self Attention Mechanism in Transformers"] idea_63cbecc0_9206_4fbd_ab74_3ddf4c14b849["Orthogonal Matrix Definition and Properties"] idea_6618d51d_5171_4594_b09a_915967624b02["Computational Graphs"] idea_6fb039a1_f730_4bc3_94d7_ff2aa2ce57f1["Eigenvalues and Spectral Radius of a Square Matrix"] idea_6fe1293c_01c7_434f_ad1b_ffba103f1548["Directional Derivative as the Gradient Dotted with a Unit Vector in Calculus"] idea_709beb06_52e4_45db_bdf3_5cd667c61872["Continuity Equation Conserves Mass as a Density Moves Through Space"] idea_7a152b50_947c_4856_9a6d_be84588b0618["Eigenvalues and Eigenvectors Definitions in Linear Algebra"] idea_814996b8_7858_4b5d_8445_5346a1da32c0["Partial Derivative"] idea_81632943_694b_4458_bd18_16a9d0d32f83["Chain Rule"] idea_82e778e7_32ba_48b7_8b9d_f85f86b9d2bb["Information Entropy"] idea_85ca84c8_641f_4d0f_a5b9_1a183122b97c["Probability Distribution"] idea_8b094e66_4362_455f_a28c_fed90bdbcd2c["Singular Value Decomposition via Eigenvalue Decomposition of A^TA and AA^T"] idea_9246e125_24f5_452a_a5ce_0d01fcaac369["Softmax as the Maximum-Entropy Distribution at Fixed Mean Energy"] idea_93b2eb6d_59cf_48c8_8f92_403240dfaf0e["Fourier Neural Operators for Learning Solution Operators of PDEs"] idea_93c0dce9_8dfd_41c5_a057_8460cba82afd["Length as Square Root of Self-Dot-Product"] idea_93d14331_7850_46dc_8b1f_b9755b66e8c6["Taylor Series Expansion of Functions via Derivatives at a Point"] idea_94355247_1d5d_4b29_ab3e_790ce585a05f["Softmax Jacobian diag(p) Minus pp^T in Backpropagation"] idea_96f7f65a_928a_4ce9_8f6f_6cefe2b2f969["Residual Connections and Layer Normalization"] idea_9d5af745_9ae4_4d63_a83d_f8b4e090e5d6["Eigenvalues and Eigenvectors for Symmetric Matrices"] idea_9d925e5b_08a0_4fff_9617_5697738d4c1c["Partial Derivatives"] idea_9dfe899a_37a6_40ae_b5ed_7c3adfa0f10d["The Transpose and Why K^T Appears in Attention"] idea_a31b45b3_0498_44c6_a5ac_13af9aed69de["Self-Attention as a Nonlocal Diffusion Step Over Tokens"] idea_a7ace413_fb9b_4e75_a012_b3e4bfbc71e2["KL Divergence"] idea_a84f8d87_29ec_4984_b0f9_f354ee23dc6c["Matrix Addition in Linear Algebra"] idea_aabf9033_0357_4123_bde6_954ff647b321["Exploding Gradient Problem"] idea_ac9190b2_ae3f_463a_bd41_cd8a752b27c2["Expected Value"] idea_ad5baa21_52d6_4945_a080_fc4600397ca1["Separation of Variables for PDEs"] idea_b1cb2d77_2815_46b1_9d1c_2334dacac6e9["Orthogonal Projections in Linear Algebra"] idea_b243628c_ef71_42ec_a2c9_beecf5bb501b["Cross-Entropy as an Optimisation Objective"] idea_b252bb59_a868_4b21_b1b6_dd8c4648fece["Partial Derivatives Multivariable Analysis"] idea_b36de16b_09ff_43ed_a231_8b001c6de60d["Dot Product Operation for Vectors in Linear Algebra"] idea_c4beb02b_7eed_40d6_8dc6_5502bbdd5faa["Logarithm"] idea_c9226813_92fb_44a2_afb4_7764850f8852["Layer Normalization Gradient Through Mean and Variance in Backpropagation"] idea_c9337ac2_d822_495d_9450_0f302bb15e3b["Loss Functions and the Gradient with Respect to Parameters"] idea_d44e3468_0a95_4f67_81e9_affdebc07908["Backpropagating Through Scaled Dot-Product Attention to Queries, Keys and Values"] idea_d5c775d7_f03f_46ce_9583_2ef6d0a14b95["Derivatives of Exponential and Logarithmic Functions in Single-Variable Calculus"] idea_d683780a_8fdd_4bc4_af85_6dfe00628de4["Cosine Similarity Between Two Vectors in Machine Learning"] idea_d6a5fd0f_b4a3_4170_84b2_c499dcde0ea5["Boltzmann Distribution"] idea_e4fe9fbf_bf5b_4fa9_84b1_a2b4db28e7ae["Stochastic Gradient Descent and the Adam Optimizer"] idea_e52b5f92_c990_4f65_b099_73b2f43624ad["Partition Function"] idea_e781a38a_a0a5_4a22_b52f_f84e8a28a085["Heat Diffusion Equation in Homogeneous Media"] idea_ec01436d_a40c_4d35_9c53_481e94f6fd76["Rank and Column Space as Information Capacity"] idea_edf4ccae_e5cb_4eb1_9a64_20d97944748d["Transformer Residual Update as One Euler Step of a Neural ODE"] idea_f4c64527_f83e_47e4_98d5_3ffc1def9fc8["Likelihood Function"] idea_f54f528b_eec3_4bce_9a6c_e315b8f03d55["Layer Normalization as a Per-Vector Operation"] idea_fa4efe45_7cde_4d2d_a7ae_f8dc459239aa["Matrix Diagonalization: A = PDP^-1 via Eigenvectors and Eigenvalues"] idea_fcb2741c_5927_4c5e_aed6_1a58d5cadac8["Positional Encoding"] idea_fe3536b4_5622_4fe7_ac62_5c788d15b2b0["Computing the Transpose Operation on Matrices in Linear Algebra"] idea_152a5426_e186_4377_bbab_6addd7f7966f --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_a31b45b3_0498_44c6_a5ac_13af9aed69de --> idea_16dd60da_c5de_4e82_9413_78dff90cfa85 idea_11c10d16_59a8_46c3_96be_9925f4fa30e6 --> idea_58ef03a7_696d_4ace_a108_26f6241d2864 idea_c9337ac2_d822_495d_9450_0f302bb15e3b --> idea_11c10d16_59a8_46c3_96be_9925f4fa30e6 idea_fe3536b4_5622_4fe7_ac62_5c788d15b2b0 --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_e52b5f92_c990_4f65_b099_73b2f43624ad --> idea_9246e125_24f5_452a_a5ce_0d01fcaac369 idea_a7ace413_fb9b_4e75_a012_b3e4bfbc71e2 --> idea_3db74e87_2793_4bad_bc9a_24bdef5ee432 idea_a31b45b3_0498_44c6_a5ac_13af9aed69de --> idea_21e80959_f5e3_4c74_9f10_bedc5b9f3504 idea_2e814f08_a86d_4308_acc1_f0419bdb9da0 --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_2fa29f8e_6b5f_44ba_8133_ade5fa6f4cf0 --> idea_58ef03a7_696d_4ace_a108_26f6241d2864 idea_3b5c9594_bb5c_4337_95b5_aabf0cdac27f --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_e781a38a_a0a5_4a22_b52f_f84e8a28a085 --> idea_250be30b_2ba3_4625_9c5c_3697ece9f37b idea_3e805dc8_1e55_4156_ac55_f363c0ff3cb1 --> idea_20feb64e_1a7d_4901_872e_c8516a8b359e idea_2132fb22_760b_47ad_a3f5_b8bea6818535 --> idea_11c10d16_59a8_46c3_96be_9925f4fa30e6 idea_425c3778_1a9c_4262_9b1f_0e41fb1a33a6 --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_42b61ac3_a40f_42bb_99ff_00bcf094e419 --> idea_5f68aa37_e831_49c9_a5ec_b8143369eca1 idea_96f7f65a_928a_4ce9_8f6f_6cefe2b2f969 --> idea_16dd60da_c5de_4e82_9413_78dff90cfa85 idea_6fb039a1_f730_4bc3_94d7_ff2aa2ce57f1 --> idea_16dd60da_c5de_4e82_9413_78dff90cfa85 idea_0dbae8aa_af97_4e8f_8c22_ea074673ad0b --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_d6a5fd0f_b4a3_4170_84b2_c499dcde0ea5 --> idea_9246e125_24f5_452a_a5ce_0d01fcaac369 idea_58ef03a7_696d_4ace_a108_26f6241d2864 --> idea_11c10d16_59a8_46c3_96be_9925f4fa30e6 idea_6618d51d_5171_4594_b09a_915967624b02 --> idea_11c10d16_59a8_46c3_96be_9925f4fa30e6 idea_3e805dc8_1e55_4156_ac55_f363c0ff3cb1 --> idea_18bd63d3_d2a3_45f1_a05f_367c4625ead4 idea_aabf9033_0357_4123_bde6_954ff647b321 --> idea_20feb64e_1a7d_4901_872e_c8516a8b359e idea_f54f528b_eec3_4bce_9a6c_e315b8f03d55 --> idea_21e80959_f5e3_4c74_9f10_bedc5b9f3504 idea_1ab16196_1504_4e2e_a184_afa8f3ed0a25 --> idea_d44e3468_0a95_4f67_81e9_affdebc07908 idea_edf4ccae_e5cb_4eb1_9a64_20d97944748d --> idea_20feb64e_1a7d_4901_872e_c8516a8b359e idea_b243628c_ef71_42ec_a2c9_beecf5bb501b --> idea_1b4c5424_3f70_45ab_b2eb_fb7af1963c4d idea_9d5af745_9ae4_4d63_a83d_f8b4e090e5d6 --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_fcb2741c_5927_4c5e_aed6_1a58d5cadac8 --> idea_5f68aa37_e831_49c9_a5ec_b8143369eca1 idea_1ab16196_1504_4e2e_a184_afa8f3ed0a25 --> idea_5f68aa37_e831_49c9_a5ec_b8143369eca1 idea_d5c775d7_f03f_46ce_9583_2ef6d0a14b95 --> idea_1b4c5424_3f70_45ab_b2eb_fb7af1963c4d idea_11c10d16_59a8_46c3_96be_9925f4fa30e6 --> idea_c9226813_92fb_44a2_afb4_7764850f8852 idea_a31b45b3_0498_44c6_a5ac_13af9aed69de --> idea_93b2eb6d_59cf_48c8_8f92_403240dfaf0e idea_aabf9033_0357_4123_bde6_954ff647b321 --> idea_18bd63d3_d2a3_45f1_a05f_367c4625ead4 idea_b243628c_ef71_42ec_a2c9_beecf5bb501b --> idea_3db74e87_2793_4bad_bc9a_24bdef5ee432 idea_4cd5b655_dc4d_4bac_8efd_a9d8eacee3a7 --> idea_93b2eb6d_59cf_48c8_8f92_403240dfaf0e idea_709beb06_52e4_45db_bdf3_5cd667c61872 --> idea_21e80959_f5e3_4c74_9f10_bedc5b9f3504 idea_a84f8d87_29ec_4984_b0f9_f354ee23dc6c --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_169779a8_f66e_4265_88af_918ed56564fe --> idea_16dd60da_c5de_4e82_9413_78dff90cfa85 idea_edf4ccae_e5cb_4eb1_9a64_20d97944748d --> idea_21e80959_f5e3_4c74_9f10_bedc5b9f3504 idea_ec01436d_a40c_4d35_9c53_481e94f6fd76 --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_0dccfd4a_bb1d_4d4d_9d7e_36cfd1509b87 --> idea_11c10d16_59a8_46c3_96be_9925f4fa30e6 idea_93c0dce9_8dfd_41c5_a057_8460cba82afd --> idea_18bd63d3_d2a3_45f1_a05f_367c4625ead4 idea_06851353_d0ff_4de8_981d_022050eac719 --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_c9337ac2_d822_495d_9450_0f302bb15e3b --> idea_58ef03a7_696d_4ace_a108_26f6241d2864 idea_94355247_1d5d_4b29_ab3e_790ce585a05f --> idea_1b4c5424_3f70_45ab_b2eb_fb7af1963c4d idea_c4beb02b_7eed_40d6_8dc6_5502bbdd5faa --> idea_3db74e87_2793_4bad_bc9a_24bdef5ee432 idea_6fe1293c_01c7_434f_ad1b_ffba103f1548 --> idea_58ef03a7_696d_4ace_a108_26f6241d2864 idea_9dfe899a_37a6_40ae_b5ed_7c3adfa0f10d --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_55daa1fa_5d15_419b_af55_c3bf09a43af9 --> idea_250be30b_2ba3_4625_9c5c_3697ece9f37b idea_58ef03a7_696d_4ace_a108_26f6241d2864 --> idea_18bd63d3_d2a3_45f1_a05f_367c4625ead4 idea_82e778e7_32ba_48b7_8b9d_f85f86b9d2bb --> idea_9246e125_24f5_452a_a5ce_0d01fcaac369 idea_81632943_694b_4458_bd18_16a9d0d32f83 --> idea_11c10d16_59a8_46c3_96be_9925f4fa30e6 idea_b1cb2d77_2815_46b1_9d1c_2334dacac6e9 --> idea_c9226813_92fb_44a2_afb4_7764850f8852 idea_3c6874e9_d0f7_442c_b1b8_08cf55e56203 --> idea_1b4c5424_3f70_45ab_b2eb_fb7af1963c4d idea_85ca84c8_641f_4d0f_a5b9_1a183122b97c --> idea_3db74e87_2793_4bad_bc9a_24bdef5ee432 idea_fa4efe45_7cde_4d2d_a7ae_f8dc459239aa --> idea_8b094e66_4362_455f_a28c_fed90bdbcd2c idea_9d925e5b_08a0_4fff_9617_5697738d4c1c --> idea_11c10d16_59a8_46c3_96be_9925f4fa30e6 idea_93c0dce9_8dfd_41c5_a057_8460cba82afd --> idea_d683780a_8fdd_4bc4_af85_6dfe00628de4 idea_2fa29f8e_6b5f_44ba_8133_ade5fa6f4cf0 --> idea_3db74e87_2793_4bad_bc9a_24bdef5ee432 idea_ad5baa21_52d6_4945_a080_fc4600397ca1 --> idea_16dd60da_c5de_4e82_9413_78dff90cfa85 idea_ac9190b2_ae3f_463a_bd41_cd8a752b27c2 --> idea_21e80959_f5e3_4c74_9f10_bedc5b9f3504 idea_f54f528b_eec3_4bce_9a6c_e315b8f03d55 --> idea_c9226813_92fb_44a2_afb4_7764850f8852 idea_5f964e9f_1680_4dd2_9522_7ffe33552992 --> idea_d44e3468_0a95_4f67_81e9_affdebc07908 idea_93d14331_7850_46dc_8b1f_b9755b66e8c6 --> idea_58ef03a7_696d_4ace_a108_26f6241d2864 idea_5f964e9f_1680_4dd2_9522_7ffe33552992 --> idea_5f68aa37_e831_49c9_a5ec_b8143369eca1 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