Experimental Materials for "C-Softmax: A Contextual-Softmax Operator Incorporating Row and Column Priorities"

This folder contains the experimental validation of all theorems for the contextual softmax (C-Softmax) operator and its bioinformatics application for Hemoglobin-β (HBB).

Kartal, Hakan Emre [email protected] 0000-0002-3952-7235 DOI: 10.36227/techrxiv.175647872.27401306/v1

📊 Comparative Summary Table: C-Softmax vs. Other Methods

The following table provides a comparative overview of C-Softmax with other Softmax-based methods from both theoretical and practical perspectives.

Method	Differentiability	Sparse Output	Context Integration	Temperature / Entropy Control	Calibration	Reparametrization	Notes
Softmax (classical)	Infinite ($$C^\infty$$)	No	No	None (unless $$\tau$$ added)	Weak	No	Basic probability projection; smooth gradient.
C-Softmax (this work)	Infinite ($$C^\infty$$)	No $$\dagger$$	Yes ($$\vec{\alpha}, \vec{\omega}, \vec{\beta}$$)	Yes (via $$\tau$$)	Strong (context-dependent)	No	Context-aware probability projection with external priorities.
Sparsemax (Martins & Astudillo, 2016)	Piecewise differentiable	Yes	No	None	Mixed	No	Euclidean projection onto simplex; interpretability advantage.
Gumbel-Softmax / Concrete (Jang et al., 2017; Maddison et al., 2017)	Infinite ($$C^\infty$$, relaxation)	No	No	Yes (via $$\tau$$)	Indirect	Yes	Enables differentiable sampling of discrete variables (VAE, RL).
Temperature Scaling (Guo et al., 2017)	Infinite ($$C^\infty$$)	No	No	Yes (post-hoc $$\tau$$)	Strong (post-hoc)	No	Post-hoc calibration; does not change the decision rule.
Label Smoothing (Szegedy et al., 2016)	Infinite ($$C^\infty$$)	No	No	Indirect (target distribution smoothing)	Typically improves	No	Reduces overconfidence; mixes targets.

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† C-Softmax provides full support; however, extreme values of context parameters (e.g., α)
may lead to near-zero probabilities, resulting in effectively sparse distributions.

📌 Notably, $$\frac{\partial L}{\partial \ln \alpha_i} = p_i - y_i$$ allows context parameters to be directly optimized within gradient-based learning frameworks.

Files

• CSoftmaxTest.dpr: It is the Pascal language implementation of the C-Softmax function in the article.
• c_softmax_experiments_2025_v1.zip:
• c_softmax_numpy_final.ipynb: Interactive Jupyter Notebook (compatible with Google Colab).
• c_softmax_numpy_final_ipynb - Colab.pdf: Results and visualizations of all tests.
• c_softmax_numpy_final.py: Standalone Python implementation.
• fig1_limit_behavior.png: C-Softmax Limit Behavior (Teorem 5) graph.
• fig2_alpha_learning.png: Adaptive Learning of α (Teorem 2) graph.
• fig3_entropy_temp.png: Entropy-Temperature Sensitivity (Teorem 7) graph.
• full_outputs.txt: All test results are here.
• HBB.zip: All the ingredients required for Hemoglobin-β (HBB).
  • P68871.fasta: This a bioinformatics file in FASTA fmat containing the Hemoglobin-β protein sequence from the UniProt (Universal Protein Resource) database.
  • last_alignment_data.txt, final_alignment_used.txt and final_alignment.clustal: This is a Hemoglobin-β protein alignment file created in Clustal format.
  • C-Softmax_HBB_Notebook_v2_en.ipynb - Colab.pdf: Results of HBB tests.
  • C_Softmax_HBB_Notebook_v2_en.ipynb: Interactive Jupyter Notebook (compatible with Google Colab).
  • blosum62.txt:

Direct Download

Installation

If you don't have the Python interpreter, get the latest Python version from here.

pip install numpy matplotlib 
python c_softmax_numpy_final.py

This work was first submitted on August 18, 2025 (TechRxiv Cases: #250826-000675, #250826-000685, #250827-000874, and #250828-000843), and proof-of-concept works were previously made publicly available via Zenodo (DOI: 10.5281/zenodo.16881843). The official preprint publication date was later recorded as August 26, 2025, due to a metadata correction.