Spend the Observation Budget on the Target: Feature Allocation for Hidden Digit Blocks
Abstract
Large variation in an observed feature does not necessarily make it useful for predicting a missing region. We compared two training-defined observation menus for four concealed Optdigits features. With sixteen visible features, target-directed greedy selection gave test MSE 0.07122, versus 0.08336 for marginal-variance selection. The advantage persisted with thirty-two observations. Both menus were fixed before the 1,797 test images were evaluated. The result supports task-specific allocation within this numerical setting, while leaving physical acquisition cost and the global optimality of the selected menu unresolved.
Train-only allocation experiment on public block-count features. Observation count is the resource unit; acquisition time, physical scanning, and a specialized inpainting model are not evaluated.
Original publication: October 31, 2026.
References
- Liu, Guilin; Reda, Fitsum A.; Shih, Kevin J.; Wang, Ting-Chun; Tao, Andrew; Catanzaro, Bryan. (2018). Image Inpainting for Irregular Holes Using Partial Convolutions. arXiv:1804.07723. https://arxiv.org/abs/1804.07723.
- Tipping, Michael E.; Bishop, Christopher M.. (1999). Probabilistic Principal Component Analysis. Journal of the Royal Statistical Society Series B: Statistical Methodology. https://doi.org/10.1111/1467-9868.00196.
- You, Jiaxuan; Ma, Xiaobai; Ding, Daisy Yi; Kochenderfer, Mykel; Leskovec, Jure. (2020). Handling Missing Data with Graph Representation Learning. arXiv:2010.16418. https://arxiv.org/abs/2010.16418.
- Krause, Andreas; Singh, Ajit; Guestrin, Carlos. (2008). Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies. Journal of Machine Learning Research,9(8),235-284. https://www.jmlr.org/papers/v9/krause08a.html.
- Alpaydin, E.; Kaynak, C. (1998). Optical Recognition of Handwritten Digits [Dataset]. UCI Machine Learning Repository. https://doi.org/10.24432/C50P49.
- White, Halbert. “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity.” Econometrica, vol. 48, no. 4, 1980, pp. 817-838. https://doi.org/10.2307/1912934.
- Suvorov, Roman; Logacheva, Elizaveta; Mashikhin, Anton; Remizova, Anastasia; Ashukha, Arsenii; Silvestrov, Aleksei; Kong, Naejin; Goka, Harshith; Park, Kiwoong; Lempitsky, Victor. (2021). Resolution-robust Large Mask Inpainting with Fourier Convolutions. arXiv:2109.07161. https://arxiv.org/abs/2109.07161.
- Yu, Jiahui; Lin, Zhe; Yang, Jimei; Shen, Xiaohui; Lu, Xin; Huang, Thomas. (2018b). Free-Form Image Inpainting with Gated Convolution. arXiv:1806.03589. https://arxiv.org/abs/1806.03589.
- Shannon, C.E.. (1949). Communication in the Presence of Noise. Proceedings of the IRE. https://doi.org/10.1109/jrproc.1949.232969.
- Wilkinson, Mark D.; Dumontier, Michel; Aalbersberg, IJsbrand Jan; Appleton, Gabrielle; Axton, Myles; Baak, Arie; Blomberg, Niklas; Boiten, Jan-Willem; da Silva Santos, Luiz Bonino; Bourne, Philip E.; Bouwman, Jildau; Brookes, Anthony J.; Clark, Tim; Crosas, Mercè; Dillo, Ingrid; Dumon, Olivier; Edmunds, Scott; Evelo, Chris T.; Finkers, Richard; Gonzalez-Beltran, Alejandra; Gray, Alasdair J.G.; Groth, Paul; Goble, Carole; Grethe, Jeffrey S.; Heringa, Jaap; ’t Hoen, Peter A.C; Hooft, Rob; Kuhn, Tobias; Kok, Ruben; Kok, Joost; Lusher, Scott J.; Martone, Maryann E.; Mons, Albert; Packer, Abel L.; Persson, Bengt; Rocca-Serra, Philippe; Roos, Marco; van Schaik, Rene; Sansone, Susanna-Assunta; Schultes, Erik; Sengstag, Thierry; Slater, Ted; Strawn, George; Swertz, Morris A.; Thompson, Mark; van der Lei, Johan; van Mulligen, Erik; Velterop, Jan; Waagmeester, Andra; Wittenburg, Peter; Wolstencroft, Katherine; Zhao, Jun; Mons, Barend. (2016). The FAIR Guiding Principles for scientific data management and stewardship. Scientific Data. https://doi.org/10.1038/sdata.2016.18.
- Carroll, Raymond J., David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu. “Measurement Error in Nonlinear Models.” 2006. https://doi.org/10.1201/9781420010138.
- Furnival, George M.; Wilson, Robert W.. (1974). Regressions by Leaps and Bounds. Technometrics. https://doi.org/10.1080/00401706.1974.10489231.
- Meng, Zijie; Zeng, Yuanze; Chang, Xiang; Xu, Tianshuo; Chao, Fei; Cao, Xixin; Shang, Changjing; Shen, Qiang. (2025). Orpaint: a zero-shot inpainting model for oracle bone inscription rubbings with visual mamba block. Science China Information Sciences, 68(8), 189102. https://doi.org/10.1007/s11432-024-4493-4.
- Virtanen, Pauli; Gommers, Ralf; Oliphant, Travis E.; Haberland, Matt; Reddy, Tyler; Cournapeau, David; Burovski, Evgeni; Peterson, Pearu; Weckesser, Warren; Bright, Jonathan; van der Walt, Stéfan J.; Brett, Matthew; Wilson, Joshua; Millman, K. Jarrod; Mayorov, Nikolay; Nelson, Andrew R. J.; Jones, Eric; Kern, Robert; Larson, Eric; Carey, C J; Polat, İlhan; Feng, Yu; Moore, Eric W.; VanderPlas, Jake; Laxalde, Denis; Perktold, Josef; Cimrman, Robert; Henriksen, Ian; Quintero, E. A.; Harris, Charles R.; Archibald, Anne M.; Ribeiro, Antônio H.; Pedregosa, Fabian; van Mulbregt, Paul; SciPy 1.0 Contributors; Vijaykumar, Aditya; Bardelli, Alessandro Pietro; Rothberg, Alex; Hilboll, Andreas; Kloeckner, Andreas; Scopatz, Anthony; Lee, Antony; Rokem, Ariel; Woods, C. Nathan; Fulton, Chad; Masson, Charles; Häggström, Christian; Fitzgerald, Clark; Nicholson, David A.; Hagen, David R.; Pasechnik, Dmitrii V.; Olivetti, Emanuele; Martin, Eric; Wieser, Eric; Silva, Fabrice; Lenders, Felix; Wilhelm, Florian; Young, G.; Price, Gavin A.; Ingold, Gert-Ludwig; Allen, Gregory E.; Lee, Gregory R.; Audren, Hervé; Probst, Irvin; Dietrich, Jörg P.; Silterra, Jacob; Webber, James T; Slavič, Janko; Nothman, Joel; Buchner, Johannes; Kulick, Johannes; Schönberger, Johannes L.; de Miranda Cardoso, José Vinícius; Reimer, Joscha; Harrington, Joseph; Rodríguez, Juan Luis Cano; Nunez-Iglesias, Juan; Kuczynski, Justin; Tritz, Kevin; Thoma, Martin; Newville, Matthew; Kümmerer, Matthias; Bolingbroke, Maximilian; Tartre, Michael; Pak, Mikhail; Smith, Nathaniel J.; Nowaczyk, Nikolai; Shebanov, Nikolay; Pavlyk, Oleksandr; Brodtkorb, Per A.; Lee, Perry; McGibbon, Robert T.; Feldbauer, Roman; Lewis, Sam; Tygier, Sam; Sievert, Scott; Vigna, Sebastiano; Peterson, Stefan; More, Surhud; Pudlik, Tadeusz; Oshima, Takuya; Pingel, Thomas J.; Robitaille, Thomas P.; Spura, Thomas; Jones, Thouis R.; Cera, Tim; Leslie, Tim; Zito, Tiziano; Krauss, Tom; Upadhyay, Utkarsh; Halchenko, Yaroslav O.; Vázquez-Baeza, Yoshiki. (2020). SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods. https://doi.org/10.1038/s41592-019-0686-2.
- Bertsimas, Dimitris; King, Angela; Mazumder, Rahul. (2016). Best subset selection via a modern optimization lens. The Annals of Statistics. https://doi.org/10.1214/15-aos1388.
- Hoerl, Arthur E.; Kennard, Robert W.. (1970). Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics. https://doi.org/10.1080/00401706.1970.10488634.
- Candes, Emmanuel J; Plan, Yaniv. (2010). Matrix Completion With Noise. Proceedings of the IEEE. https://doi.org/10.1109/jproc.2009.2035722.
- Harris, Charles R.; Millman, K. Jarrod; van der Walt, Stéfan J.; Gommers, Ralf; Virtanen, Pauli; Cournapeau, David; Wieser, Eric; Taylor, Julian; Berg, Sebastian; Smith, Nathaniel J.; Kern, Robert; Picus, Matti; Hoyer, Stephan; van Kerkwijk, Marten H.; Brett, Matthew; Haldane, Allan; del Río, Jaime Fernández; Wiebe, Mark; Peterson, Pearu; Gérard-Marchant, Pierre; Sheppard, Kevin; Reddy, Tyler; Weckesser, Warren; Abbasi, Hameer; Gohlke, Christoph; Oliphant, Travis E.. (2020). Array programming with NumPy. Nature. https://doi.org/10.1038/s41586-020-2649-2.
- Cao, Wei; Wang, Dong; Li, Jian; Zhou, Hao; Li, Lei; Li, Yitan. (2018). BRITS: Bidirectional Recurrent Imputation for Time Series. arXiv:1805.10572. https://arxiv.org/abs/1805.10572.
- Cook, R. Dennis. (1977). Detection of Influential Observation in Linear Regression. Technometrics. https://doi.org/10.1080/00401706.1977.10489493.
- Joshi, S.; Boyd, S.. (2009). Sensor Selection via Convex Optimization. IEEE Transactions on Signal Processing. https://doi.org/10.1109/tsp.2008.2007095.
- Vincent, Pascal; Larochelle, Hugo; Bengio, Yoshua; Manzagol, Pierre-Antoine. (2008). Extracting and composing robust features with denoising autoencoders. Proceedings of the 25th international conference on Machine learning - ICML '08. https://doi.org/10.1145/1390156.1390294.
- Cameron, A. Colin, and Douglas L. Miller. “A Practitioner’s Guide to Cluster-Robust Inference.” Journal of Human Resources, vol. 50, no. 2, 2015, pp. 317-372. https://doi.org/10.3368/jhr.50.2.317.
- Pathak, Deepak; Krahenbuhl, Philipp; Donahue, Jeff; Darrell, Trevor; Efros, Alexei A.. (2016). Context Encoders: Feature Learning by Inpainting. arXiv:1604.07379. https://arxiv.org/abs/1604.07379.
- Cini, Andrea; Marisca, Ivan; Alippi, Cesare. (2021). Filling the G_ap_s: Multivariate Time Series Imputation by Graph Neural Networks. arXiv:2108.00298. https://arxiv.org/abs/2108.00298.
- Nazeri, Kamyar; Ng, Eric; Joseph, Tony; Qureshi, Faisal Z.; Ebrahimi, Mehran. (2019). EdgeConnect: Generative Image Inpainting with Adversarial Edge Learning. arXiv:1901.00212. https://arxiv.org/abs/1901.00212.
- Mazumder, Rahul; Hastie, Trevor; Tibshirani, Robert. (2010). Spectral Regularization Algorithms for Learning Large Incomplete Matrices. Journal of Machine Learning Research 11, 2287-2322. https://jmlr.org/papers/v11/mazumder10a.html.
- Yoon, Jinsung; Jordon, James; van der Schaar, Mihaela. (2018). GAIN: Missing Data Imputation using Generative Adversarial Nets. arXiv:1806.02920. https://arxiv.org/abs/1806.02920.
- Pearson, Karl. (1901). LIII. On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. https://doi.org/10.1080/14786440109462720.
- Gebru, Timnit, et al. "Datasheets for datasets" Communications of the ACM, 2021. https://doi.org/10.1145/3458723.
Published
Issue
Section
License
Copyright (c) 2026 Brandon Franklin, Stephen Young, Ryan King

This work is licensed under a Creative Commons Attribution 4.0 International License.