pith. machine review for the scientific record. sign in

arxiv: 2604.03263 · v1 · submitted 2026-03-12 · 💻 cs.CL · cs.AI· cs.NE

Recognition: 2 theorem links

· Lean Theorem

LPC-SM: Local Predictive Coding and Sparse Memory for Long-Context Language Modeling

Keqin Xie
Authors on Pith no claims yet

Pith reviewed 2026-05-15 11:19 UTC · model grok-4.3

classification 💻 cs.CL cs.AIcs.NE
keywords long-context language modelinghybrid architecturepredictive codingsparse memoryautoregressive modelingorthogonal novelty transportattention alternatives
0
0 comments X

The pith

Long-context autoregressive modeling can be organized around a broader division of labor than attention alone by separating local attention, persistent memory, predictive correction, and run-time control within each block.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces LPC-SM as a hybrid architecture that assigns local attention to immediate token interactions, persistent memory to long-range state, predictive correction to ongoing adjustments, and run-time control to adaptive decisions, all managed inside the same block. Orthogonal Novelty Transport governs writes to the slow memory component. On a 158M-parameter model, the design lowers language-modeling loss across base training, mathematical continuation, and 4096-token sequences while improving a delayed-identifier diagnostic, showing that attention need not carry every aspect of long-range dependency.

Core claim

LPC-SM separates local attention, persistent memory, predictive correction, and run-time control within the same block and uses Orthogonal Novelty Transport to govern slow-memory writes, yielding stable training at sequence length 4096 with final LM loss 11.582 and cross-entropy improvement on the delayed-identifier task from 14.396 to 12.031.

What carries the argument

LPC-SM block that decomposes sequence modeling into local attention, persistent memory, predictive correction, and run-time control, with Orthogonal Novelty Transport controlling slow-memory writes.

If this is right

  • Removing the mHC component raises Stage-A final LM loss from 12.630 to 15.127.
  • Adaptive sparse control lowers Stage-B final LM loss from 12.137 to 10.787 relative to fixed-ratio continuation.
  • The full model stays stable through 4096 tokens and ends Stage C at LM loss 11.582.
  • The delayed-identifier diagnostic improves from 14.396 to 12.031 in cross-entropy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same component split could be tested in models larger than 158M to check whether the loss reductions continue to scale.
  • Orthogonal Novelty Transport might be ported to other memory-augmented transformers to isolate its contribution from the rest of the LPC-SM block.
  • Running the architecture at 8k or 16k tokens could reveal whether the explicit separation prevents the quadratic bottlenecks that appear in pure attention models.
  • If the four roles can be re-wired at inference time, the design might support task-specific routing without retraining the entire network.

Load-bearing premise

The separation of local attention, persistent memory, predictive correction, and run-time control governed by Orthogonal Novelty Transport produces stable gains at length 4096 without hidden instabilities or costs.

What would settle it

Observing rising loss or new instabilities when the same 158M model is trained at lengths substantially beyond 4096 would falsify the claim of stable improvement from this division of labor.

Figures

Figures reproduced from arXiv: 2604.03263 by Keqin Xie.

Figure 1
Figure 1. Figure 1: Overall LPC-SM stack. 3.1 Block Structure Each block begins with RMSNorm [28] and then combines three information sources at token position 𝑡: a local-attention read, a dual-timescale memory read, and a predictive correction. The local-attention path is windowed and causal, 𝑎𝑡 = Attn(ℎmax(1,𝑡−𝑤+1):𝑡), where 𝑤 is the local window. The goal of this path is local precision rather than long-range storage. In t… view at source ↗
Figure 2
Figure 2. Figure 2: A single LPC-SM block. 3.2 Dual-Timescale Memory and ONT At the end of chunk 𝐶𝑘, the block forms a chunk summary 𝑐𝑘 = 1 |𝐶𝑘 | ∑︁ 𝑡 ∈𝐶𝑘 𝑚 𝑓 𝑡 . The slow-memory gate is input dependent, 𝑔𝑘 = 𝜎(𝑊𝑔ℎ𝑡). The question is how 𝑐𝑘 should be transported before it is written into the slow state. ONT defines the aligned component relative to the previous slow state 𝑚 𝑠 𝑘−1 by proj(𝑐𝑘 | 𝑚 𝑠 𝑘−1 ) =    0, if ∥… view at source ↗
Figure 3
Figure 3. Figure 3: Orthogonal Novelty Transport for slow-memory writes. 3.3 Correction and Stopping The block also predicts the current hidden state from local context and memory and then corrects that prediction with an explicit mismatch signal. The predictor is initialized by ℎˆ (0) 𝑡 = 𝑓pred [𝑎𝑡 ∥ 𝑟𝑡]  , and refined iteratively: ℎˆ (𝑠+1) 𝑡 = ℎˆ (𝑠) 𝑡 + 𝑓refine [𝑎𝑡 ∥ 𝑟𝑡 ∥ ℎ𝑡 − ℎˆ (𝑠) 𝑡 ]  . LPC-SM does not force this pat… view at source ↗
read the original abstract

Most current long-context language models still rely on attention to handle both local interaction and long-range state, which leaves relatively little room to test alternative decompositions of sequence modeling. We propose LPC-SM, a hybrid autoregressive architecture that separates local attention, persistent memory, predictive correction, and run-time control within the same block, and we use Orthogonal Novelty Transport (ONT) to govern slow-memory writes. We evaluate a 158M-parameter model in three stages spanning base language modeling, mathematical continuation, and 4096-token continuation. Removing mHC raises the Stage-A final LM loss from 12.630 to 15.127, while adaptive sparse control improves the Stage-B final LM loss from 12.137 to 10.787 relative to a matched fixed-ratio continuation. The full route remains stable at sequence length 4096, where Stage C ends with final LM loss 11.582 and improves the delayed-identifier diagnostic from 14.396 to 12.031 in key cross-entropy. Taken together, these results show that long-context autoregressive modeling can be organized around a broader division of labor than attention alone.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes LPC-SM, a hybrid autoregressive architecture that separates local attention, persistent memory, predictive correction, and run-time control within the same block, governed by Orthogonal Novelty Transport (ONT) for slow-memory writes. It evaluates a 158M-parameter model across three training stages (base LM, mathematical continuation, 4096-token continuation), reporting that removing mHC raises Stage-A LM loss from 12.630 to 15.127, adaptive sparse control lowers Stage-B loss from 12.137 to 10.787, and the full model reaches 11.582 LM loss with 12.031 delayed-identifier cross-entropy at 4096 tokens.

Significance. If the architecture proves stable and competitive, the work would support the viability of decomposing long-context modeling into a broader division of labor than attention alone, potentially informing more modular designs for extended sequences. The staged evaluation and internal ablations provide concrete empirical anchors, but the lack of external baselines prevents assessing whether these losses reflect genuine progress.

major comments (2)
  1. [Abstract] Abstract: the central claim that long-context autoregressive modeling can be organized around a broader division of labor than attention alone is unsupported, as no matched Transformer baseline (same 158M parameters, same training stages, same 4096 evaluation length) is reported; all quantitative results are internal ablations only.
  2. [Stage C evaluation] Stage C evaluation: stability at sequence length 4096 is asserted via final LM loss 11.582, but no metrics on memory footprint, wall-clock overhead, or divergence indicators are provided relative to attention, leaving the 'no hidden costs' assumption untested.
minor comments (2)
  1. [Abstract] Abstract: loss numbers are given without error bars, statistical tests, or explicit definition of the 'matched fixed-ratio continuation' baseline.
  2. [Method] The definition and orthogonality properties of Orthogonal Novelty Transport (ONT) require expanded pseudocode or equations for reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback on the strength of our claims and the need for additional evaluation details. We respond to each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that long-context autoregressive modeling can be organized around a broader division of labor than attention alone is unsupported, as no matched Transformer baseline (same 158M parameters, same training stages, same 4096 evaluation length) is reported; all quantitative results are internal ablations only.

    Authors: We agree that the central claim would be stronger with a matched Transformer baseline under identical conditions. Our results are limited to internal ablations that isolate the contributions of mHC, adaptive sparse control, and ONT, showing loss reductions attributable to these components. We will revise the abstract and discussion sections to qualify the claim as demonstrating the viability of the proposed decomposition via these ablations, rather than asserting superiority over attention-based models. This is a partial revision since new baseline experiments cannot be added. revision: partial

  2. Referee: [Stage C evaluation] Stage C evaluation: stability at sequence length 4096 is asserted via final LM loss 11.582, but no metrics on memory footprint, wall-clock overhead, or divergence indicators are provided relative to attention, leaving the 'no hidden costs' assumption untested.

    Authors: We concur that efficiency metrics are necessary to fully substantiate the stability claim at 4096 tokens. The current manuscript uses final LM loss and the delayed-identifier cross-entropy as primary indicators. In the revision we will add memory footprint details, any observed divergence indicators from training logs, and wall-clock measurements from the existing experimental runs to address the untested assumption directly. revision: yes

standing simulated objections not resolved
  • No matched Transformer baseline with the same 158M parameters, training stages, and 4096-token evaluation was performed in the original experiments.

Circularity Check

0 steps flagged

No circularity: empirical ablations and staged losses are independent of the target claim

full rationale

The paper's chain consists of an architectural proposal (local attention + persistent memory + predictive correction + ONT-governed writes), followed by three-stage training and internal ablations (mHC removal, fixed vs. adaptive sparse control). Reported quantities are direct training losses and a delayed-identifier cross-entropy metric; none are obtained by fitting a parameter to the final claim and then re-deriving the claim from that parameter. No self-citation load-bearing step, no uniqueness theorem imported from prior work, and no renaming of known results as new organization. The derivation remains self-contained against external benchmarks because the quantitative support is generated by running the model rather than by algebraic rearrangement of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Based on abstract only; central claim rests on empirical loss improvements from component ablations, with ONT introduced as a new governing mechanism for memory writes.

invented entities (1)
  • Orthogonal Novelty Transport (ONT) no independent evidence
    purpose: To govern slow-memory writes in the hybrid architecture
    New mechanism postulated to control persistent memory updates, with no independent evidence provided in abstract.

pith-pipeline@v0.9.0 · 5500 in / 1159 out tokens · 39225 ms · 2026-05-15T11:19:28.214340+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 19 internal anchors

  1. [1]

    Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality

    Albert Gu and Tri Dao. Transformers are SSMs: Generalized models and efficient algorithms through structured state space duality, 2024. URLhttps://arxiv.org/abs/2405.21060. arXiv:2405.21060

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  2. [2]

    Griffin: Mixing Gated Linear Recurrences with Local Attention for Efficient Language Models

    Soham De, Sam Smith, Anushan Fernando, Aleksandar Botev, George Tucker, Michal Valko, Razvan Pascanu, Sebastian Ruder, Yee Whye Teh, and Donald Metzler. Griffin: Mixing gated linear recurrences with local attention for efficient language models, 2024. URLhttps://arxiv.org/abs/2402.19427. arXiv:2402.19427

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  3. [3]

    Titans: Learning to Memorize at Test Time

    Ali Behrouz, Peilin Zhong, and Vahab Mirrokni. Titans: Learning to memorize at test time, 2025. URL https://arxiv.org/abs/2501.00663. arXiv:2501.00663

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  4. [4]

    Hymba: A hybrid-head architecture for small language models, 2024

    Xin Dong, Tianyu Liu, Yuhang Zang, Yanyan Zhao, Jiapeng Zhang, Zhaopeng Tu, Chengqiang Huang, Huadong Wang, and Jie Zhou. Hymba: A hybrid-head architecture for small language models, 2024. URLhttps: //arxiv.org/abs/2411.13676. arXiv:2411.13676

    work page arXiv 2024

  5. [5]

    LongRoPE: Extending LLM Context Window Beyond 2 Million Tokens

    Yiran Ding, Li Dong, Peiyuan Liu, Kaixiong Zhou, Ermo Hua, Song Lin, Zhuang Li, Yuejie Zhang, Yuhang Cao, Lei Shang, Xin Jiang, and Qun Liu. Longrope: Extending LLM context window beyond 2 million tokens, 2024. URLhttps://arxiv.org/abs/2402.13753. arXiv:2402.13753

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  6. [6]

    InfLLM:Training-freelong-contextextrapolationfor LLMs with an efficient context memory, 2024

    Chaojun Xiao, Longyue Wang, Yingjia Wan, Yang Wang, Yuxuan Peng, Hao Zhu, Tianyu Liu, Xingyao Wang, YusenZhang, ChaojieZhang, ZhiyuanLiu, andMaosongSun. InfLLM:Training-freelong-contextextrapolationfor LLMs with an efficient context memory, 2024. URLhttps://arxiv.org/abs/2402.04617. arXiv:2402.04617

    work page arXiv 2024

  7. [7]

    Kimi Linear: An Expressive, Efficient Attention Architecture

    Yu Zhang, Yifan Chen, Yichen Gong, Zhenyu Yang, Xuanjing Huang, and Kimi Team. Kimi linear: An expressive, efficient attention architecture, 2025. URLhttps://arxiv.org/abs/2510.26692. arXiv:2510.26692

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  8. [8]

    Attention Is All You Need

    AshishVaswani,NoamShazeer,NikiParmar,JakobUszkoreit,LlionJones,AidanN.Gomez,ŁukaszKaiser,andIllia Polosukhin. Attention is all you need, 2017. URLhttps://arxiv.org/abs/1706.03762. arXiv:1706.03762

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  9. [9]

    Generating long sequences with sparse transformers,

    Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse transformers,

    work page

  10. [10]

    Generating Long Sequences with Sparse Transformers

    URLhttps://arxiv.org/abs/1904.10509. arXiv:1904.10509

    work page internal anchor Pith review Pith/arXiv arXiv 1904

  11. [11]

    Adaptive attention span in transformers, 2019

    Sainbayar Sukhbaatar, Edouard Grave, Guillaume Lample, Herve Jegou, and Armand Joulin. Adaptive attention span in transformers, 2019. URLhttps://arxiv.org/abs/1905.07799. arXiv:1905.07799

    work page arXiv 2019

  12. [12]

    Longformer: The Long-Document Transformer

    Iz Beltagy, Matthew E. Peters, and Arman Cohan. Longformer: The long-document transformer, 2020. URL https://arxiv.org/abs/2004.05150. arXiv:2004.05150

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  13. [13]

    Big bird: Transformers for longer sequences, 2020

    Manzil Zaheer, Guru Guruganesh, Avinava Dubey, Joshua Ainslie, Chris Alberti, Santiago Ontanon, Philip Pham, Anirudh Ravula, Qifan Wang, Li Yang, and Amr Ahmed. Big bird: Transformers for longer sequences, 2020. URL https://arxiv.org/abs/2007.14062. arXiv:2007.14062

    work page arXiv 2020

  14. [14]

    Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context

    Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc V. Le, and Ruslan Salakhutdinov. Transformer-XL: Attentive language models beyond a fixed-length context, 2019. URLhttps://arxiv.org/abs/1901.02860. arXiv:1901.02860

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  15. [15]

    W., Potapenko, A., Jayakumar, S

    Jack W. Rae, Anna Potapenko, Siddhant M. Jayakumar, Chloe Hillier, and Timothy P. Lillicrap. Compres- sive transformers for long-range sequence modelling, 2019. URL https://arxiv.org/abs/1911.05507. arXiv:1911.05507

    work page arXiv 2019

  16. [16]

    Aydar Bulatov, Yuri Kuratov, and Mikhail S. Burtsev. Recurrent memory transformer. InAdvances in Neural Information Processing Systems 35, 2022. doi: 10.52202/068431-0805. URLhttps://doi.org/10.52202/ 068431-0805

    work page doi:10.52202/068431-0805 2022

  17. [17]

    Retentive Network: A Successor to Transformer for Large Language Models

    Yutao Sun et al. Retentive network: A successor to transformer for large language models, 2023. URL https://arxiv.org/abs/2307.08621. arXiv:2307.08621

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  18. [18]

    RWKV: Reinventing RNNs for the transformer era

    Bo Peng et al. RWKV: Reinventing RNNs for the transformer era. InFindings of the Association for Computational Linguistics: EMNLP 2023, 2023. doi: 10.18653/v1/2023.findings-emnlp.936. URL https://doi.org/10. 18653/v1/2023.findings-emnlp.936. 18

    work page doi:10.18653/v1/2023.findings-emnlp.936 2023

  19. [19]

    Mamba: Linear-Time Sequence Modeling with Selective State Spaces

    Albert Gu and Tri Dao. Mamba: Linear-time sequence modeling with selective state spaces, 2023. URL https://arxiv.org/abs/2312.00752. arXiv:2312.00752

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  20. [20]

    Rajesh P. N. Rao and Dana H. Ballard. Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects.Nature Neuroscience, 2(1):79–87, 1999. doi: 10.1038/4580. URL https://doi.org/10.1038/4580

    work page doi:10.1038/4580 1999

  21. [21]

    A theory of cortical responses.Philosophical Transactions of the Royal Society B: Biological Sciences, 360(1456):815–836, 2005

    Karl Friston. A theory of cortical responses.Philosophical Transactions of the Royal Society B: Biological Sciences, 360(1456):815–836, 2005. doi: 10.1098/rstb.2005.1622. URLhttps://doi.org/10.1098/rstb.2005.1622

    work page doi:10.1098/rstb.2005.1622 2005

  22. [22]

    Deep Predictive Coding Networks for Video Prediction and Unsupervised Learning

    William Lotter, Gabriel Kreiman, and David Cox. Deep predictive coding networks for video prediction and unsupervised learning, 2016. URLhttps://arxiv.org/abs/1605.08104. arXiv:1605.08104

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  23. [23]

    Predictive coding approximates backprop along arbitrary computation graphs,

    Tim Whittington and Rafal Bogacz. Predictive coding approximates backprop along arbitrary computation graphs,

    work page

  24. [24]

    arXiv:2006.04182

    URLhttps://arxiv.org/abs/2006.04182. arXiv:2006.04182

    work page arXiv 2006

  25. [25]

    Adaptive Computation Time for Recurrent Neural Networks

    Alex Graves. Adaptive computation time for recurrent neural networks, 2016. URLhttps://arxiv.org/abs/ 1603.08983. arXiv:1603.08983

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  26. [26]

    Universal transformers,

    Mostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Łukasz Kaiser. Universal transformers,

    work page

  27. [27]

    Universal Transformers

    URLhttps://arxiv.org/abs/1807.03819. arXiv:1807.03819

    work page internal anchor Pith review Pith/arXiv arXiv

  28. [28]

    Depth-adaptive transformer, 2019

    Mher Elbayad, Jiatao Gu, Edouard Grave, and Michael Auli. Depth-adaptive transformer, 2019. URLhttps: //arxiv.org/abs/1910.10073. arXiv:1910.10073

    work page arXiv 2019

  29. [29]

    Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer

    Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc V. Le, Geoffrey Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-of-experts layer, 2017. URLhttps: //arxiv.org/abs/1701.06538. arXiv:1701.06538

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  30. [30]

    Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity

    William Fedus, Barret Zoph, and Noam Shazeer. Switch transformers: Scaling to trillion parameter models with simple and efficient sparsity, 2021. URLhttps://arxiv.org/abs/2101.03961. arXiv:2101.03961

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  31. [31]

    Root mean square layer normalization

    Biao Zhang and Rico Sennrich. Root mean square layer normalization. InAdvances in Neural Informa- tion Processing Systems 32 (NeurIPS), 2019. URLhttps://proceedings.neurips.cc/paper/2019/hash/ 1e8a19426224ca89e83cef47f1e7f53b-Abstract.html

    work page 2019

  32. [32]

    mHC: Manifold-Constrained Hyper-Connections

    Zhenda Xie, Yixuan Wei, Huanqi Cao, Chenggang Zhao, Chengqi Deng, Jiashi Li, Damai Dai, Huazuo Gao, Jiang Chang, Kuai Yu, Liang Zhao, Shangyan Zhou, Zhean Xu, Zhengyan Zhang, Wangding Zeng, Shengding Hu, Yuqing Wang, Jingyang Yuan, Lean Wang, and Wenfeng Liang. mHC: Manifold-constrained hyper-connections, 2025. URL https://arxiv.org/abs/2512.24880. arXiv:...

    work page internal anchor Pith review arXiv 2025

  33. [33]

    Scaling Laws for Neural Language Models

    Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language models, 2020. URLhttps: //arxiv.org/abs/2001.08361. arXiv:2001.08361

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  34. [34]

    Training Compute-Optimal Large Language Models

    Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, Tom Hennigan, Eric Noland, Katie Millican, George van den Driessche, Bogdan Damoc, Aurelia Guy, Simon Osindero, Karen Simonyan, Erich Elsen, Jack W. Rae, Oriol Vinyals, and Laurent Sifre...

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  35. [35]

    Meyer.Matrix Analysis and Applied Linear Algebra

    Carl D. Meyer.Matrix Analysis and Applied Linear Algebra. SIAM, 2000. Chapter 5: Norms, Inner Products, and Orthogonality

    work page 2000

  36. [36]

    Goldstein

    Ward Cheney and Allen A. Goldstein. Proximity maps for convex sets.Proceedings of the American Mathematical Society, 10(3):448–450, 1959. doi: 10.1090/S0002-9939-1959-0105008-8. URLhttps://doi.org/10.1090/ S0002-9939-1959-0105008-8

    work page doi:10.1090/s0002-9939-1959-0105008-8 1959

  37. [37]

    Bauschke and Jonathan M

    Heinz H. Bauschke and Jonathan M. Borwein. On projection algorithms for solving convex feasibility problems. SIAM Review, 38(3):367–426, 1996. doi: 10.1137/S0036144593251710. URLhttps://doi.org/10.1137/ S0036144593251710. 19

    work page doi:10.1137/s0036144593251710 1996

  38. [38]

    The method of alternating orthogonal projections

    Frank Deutsch. The method of alternating orthogonal projections. InApproximation Theory, Spline Functions and Applications. Springer, 1992. doi: 10.1007/978-94-011-2634-2_5. URL https://doi.org/10.1007/ 978-94-011-2634-2_5

    work page doi:10.1007/978-94-011-2634-2_5 1992

  39. [39]

    Bauschke, Hui Ouyang, and Xianfu Wang

    Heinz H. Bauschke, Hui Ouyang, and Xianfu Wang. Best approximation mappings in hilbert spaces.Mathematical Programming, 189(1):1–35, 2021. doi: 10.1007/S10107-021-01718-Y. URLhttps://doi.org/10.1007/ S10107-021-01718-Y

    work page doi:10.1007/s10107-021-01718-y 2021

  40. [40]

    Bauschke and Valentin R

    Heinz H. Bauschke and Valentin R. Koch. Projection methods: Swiss army knives for solving feasibility and best approximation problems with halfspaces. InApproximation, Optimization, and Mathematical Economics. American Mathematical Society, 2015. doi: 10.1090/CONM/636/12726. URLhttps://doi.org/10.1090/CONM/636/ 12726. 20

    work page doi:10.1090/conm/636/12726 2015