HarmonRank: Dual Approaches in Ranking

• HarmonRank is defined by two distinct methodologies: one for multi-objective live-streaming recommendations using soft-sorted, attention-driven ensembles, and the other for inferring network hierarchies via energy minimization.
• The e-commerce approach employs differentiable ranking loss with SoftSort and relation-aware self- and cross-attention mechanisms to optimize AUC and improve online engagement metrics.
• The network-based method, known as SpringRank, minimizes quadratic energy through sparse linear systems to accurately recover continuous hierarchical rankings in directed graphs.

HarmonRank refers to two distinct but prominent methodologies in the ranking literature: (1) a ranking-aligned, multi-objective ensemble for live-streaming e-commerce recommendation focusing on both inter-objective and ranking metric alignment (Xia et al., 6 Jan 2026); and (2) a physically-inspired model for hierarchical ranking in directed networks, which minimizes quadratic energy to infer continuous scores, commonly known as SpringRank (Bacco et al., 2017). Both approaches share core design philosophies of real-valued score inference and global structure exploitation, but their domains, objectives, and mathematical machinery are highly specialized and non-overlapping.

1. Multi-objective Ensemble HarmonRank in Live-streaming E-commerce Recommendation

HarmonRank, as formulated in the context of live-streaming e-commerce, addresses the problem of aggregating multiple, potentially interdependent objectives—such as purchase, comment, like, and follow probabilities—into a single ranking score. This formulation explicitly confronts the inadequacy of standard binary cross-entropy (BCE) supervision, which is misaligned with pairwise or rank-based metrics such as AUC, and fails to capture objective correlations necessary for long-term ecological balance in live-streamer-focused platforms (Xia et al., 6 Jan 2026).

1.1 Problem and Baseline Paradigms

Let $\mathbf{e}_i \in \mathbb{R}^M$ denote $M$ precomputed objective scores (e.g., purchase-probability, like-probability) for item $I_i$, and $\mathbf{y}_i \in \{0,1\}^M$ be the observed binary outcome labels per objective. Existing ensemble models learn a scalar score via $$s_i = \mathcal{F}(\mathbf{e}_i, \mathbf{u}, \mathbf{c}; \theta)$$ and optimize:

$$L_{\mathrm{BCE}}(\theta) = \sum_{m=1}^M \sum_{i=1}^N \left[ -y_i^{(m)} \log \sigma(s_i) - (1-y_i^{(m)}) \log (1-\sigma(s_i)) \right]$$

This is provably misaligned with AUC-based ranking objectives, as BCE is minimized also by trivial or non-discriminative solutions that fail to maximize AUC.

1.2 AUC-maximization via Differentiable Surrogates

HarmonRank reframes AUC as a rank-sum:

$$\mathrm{AUC}_m = \frac{1}{|\mathcal{D}_m^+||\mathcal{D}_m^-|} \sum_{i\in\mathcal{D}_m^+} \sum_{j\in\mathcal{D}_m^-} \mathbf{1}(s_i > s_j)$$

which, up to constants, is equivalent to maximizing $\langle \mathbf{r}, \mathbf{y}^{(m)} \rangle$ where $\mathbf{r}$ is the vector of discrete ranks. To yield a differentiable objective, the (hard) rank operator is replaced by SoftSort, the continuous relaxation defined as:

$$\mathrm{SoftSort}_\tau(\mathbf{s}) = \arg\min_{\mathbf{t} \in \mathcal{P}(\mathbf{r})} \frac{1}{2}\|\mathbf{t} - \mathbf{s}\|^2$$

The ranking loss per objective is defined as $$L_{\mathrm{rank}}^{(m)} = -\langle \mathrm{SoftSort}_\tau(\mathbf{s}), \mathbf{y}^{(m)} \rangle$$ and the total loss is summed over objectives.

1.3 Relation-aware Two-step Ensemble

Distinct from naïve weighted fusion, HarmonRank employs a two-step scheme:

• Step 1 (Self- and Cross-Attention): Score embeddings from all objectives are passed through a self-attention block to learn $\mathbf{A}_r \in \mathbb{R}^{M\times M}$, modeling inter-objective relations. Personalized context is injected by a cross-attention mechanism combining user and context embedding with the self-attended scores.
• Step 2 (Aggregation Paths): Parallel modules perform relation-aware fusion (from attention), relation-agnostic gating on original embeddings, and a linear path. The outputs $s_1$, $s_2$, $s_3$ are summed: $f(\mathbf{x}; \theta) = s_1 + s_2 + s_3$.
• Alignment Regularization: A penalty term enforces consistency between learned attention weights $\mathbf{A}_r$ and empirical Spearman correlations $\rho_{mn}$ among target labels.

1.4 Algorithmic Form and Implementation

HarmonRank is trained by minibatch SGD on the joint loss:

$$\mathcal{L}(\theta) = \sum_{m=1}^M L_{\mathrm{rank}}^{(m)} + \lambda \sum_{m\neq n} (A_r^{mn} - \rho_{mn})^2$$

Key implementation details include:

• Discrete embedding of scores to enhance nonlinear modeling.
• Large batch sizes ($\sim 10^4$) to stabilize SoftSort gradients.
• Fused CUDA kernels to achieve $O(N\log N)$ soft-sorting and attention during training.

1.5 Empirical Results and Comparison

HarmonRank achieves superior summed AUC and multi-metric online gains relative to BCE, label-aggregation, and RL-based baselines. Ablation studies attribute these improvements to SoftSort-based loss and explicit relational modeling. Pareto frontier analysis indicates strict improvement in trade-off efficiency over competing approaches for purchase/comment/follow objectives. HarmonRank is deployed in high-volume industrial recommendation scenarios (e.g., Kuaishou) and demonstrates robust offline and online performance across large datasets (Xia et al., 6 Jan 2026).

2. HarmonRank/SpringRank for Efficient Ranking in Networks

HarmonRank, also known as SpringRank, is a general-purpose methodology to infer continuous, real-valued hierarchical rankings in directed networks by mapping the ranking problem to an energy minimization in a system of oriented springs (Bacco et al., 2017).

2.1 Physical Model and Energy Formulation

Each directed edge $i\to j$ is modeled as an oriented spring of rest-length 1, with potential energy:

$E_{ij} = \frac{1}{2} (r_i - r_j - 1)^2$

Summing over edge weights $A_{ij}$ yields the total energy:

$$E(r) = \frac{1}{2} \sum_{i,j} A_{ij} (r_i - r_j - 1)^2$$

Minimizing $E(r)$ aligns observed directed edges with the constraint that higher-ranked nodes should be separated by a unit gap per edge.

2.2 Solution via Linear System

The stationarity condition $\frac{\partial E}{\partial r_i} = 0$ leads to a sparse linear system:

$L r = b$

where:

• $$L = D^{\mathrm{out}} + D^{\mathrm{in}} - (A + A^\top)$$, with $D^{\mathrm{out}}$ and $D^{\mathrm{in}}$ as diagonal out/in-degree matrices.
• $b = (D^{\mathrm{out}} - D^{\mathrm{in}})\mathbf{1}$.

The system is symmetric positive semidefinite, with translational degeneracy resolved by pinning one node or adding a small $\ell_2$ regularization (i.e., $E_\alpha(r) = E(r) + \frac{\alpha}{2}\|r\|^2$).

2.3 Statistical Testing and Generative Interpretations

The minimized energy $E(r^*)$ per edge is a statistic for significance of inferred hierarchy. Under the null (randomized edge directions), permutation tests yield $p$-values for the observed order structure. The physical analogy extends to a maximum-likelihood generative model where

$$\mathbb{E}[A_{ij}] = c \exp\left[ -\frac{\beta}{2}(r_i - r_j - 1)^2 \right]$$

and the likelihood is maximized over $r$ and inverse temperature $\beta$.

Conditional edge directions and probabilities are given by:

$$P_{ij}(\beta) = \frac{1}{1 + e^{-2\beta(r_i - r_j)}}$$

with $\beta$ tuned to maximize either prediction accuracy or likelihood.

2.4 Empirical Performance and Computational Aspects

SpringRank consistently recovers ground-truth hierarchies on synthetic data and outperforms classical models (Bradley-Terry-Luce, SerialRank, MVR) and ordinal methods on real datasets from diverse domains (faculty hiring, animal hierarchies, social support, sports). Statistical power is demonstrated by significantly lower $p$-values in real graphs than nulls. For networks up to $10^6$ nodes/edges, linear systems are solved in seconds using conjugate gradient with appropriate preconditioners. Code resources and detailed practical guidance are publicly available.

2.5 Implementation Considerations

To solve $L r = b$ efficiently, precompute in- and out-degrees and assemble $L$ as a sparse matrix. For $\alpha=0$, pin one node to break translational degeneracy; for large-scale graphs use multigrid or incomplete-Cholesky preconditioning. Regularization may stabilize solution where degree heterogeneity is extreme, but is unnecessary for moderately connected graphs. Comparative baselines should be regularized for fair evaluation.

3. Comparative Overview

HarmonRank (E-commerce, (Xia et al., 6 Jan 2026))	HarmonRank/SpringRank (Networks, (Bacco et al., 2017))
Multi-objective ranking ensemble; end-to-end differentiable AUC maximization; explicit relation modeling via attention and personalized cross-attention	Energy-minimization over oriented springs; efficient continuous hierarchy inference in directed networks; statistical validation via permutation and generative modeling
Task: user-item recommendation in live-streaming with multiple supervised behaviors	Task: node ranking in arbitrary directed graphs (social, biological, infrastructure, etc.)
Core method: differentiable soft-sorting, two-step "align then fuse" relational attention, SGD	Core method: linear system solution for quadratic energy; conjugate gradient or sparse direct solvers
Key metrics: AUC-sum, online engagement rates	Key metrics: energy per edge, directional accuracy, Spearman correlation

4. Scope and Impact of Applications

The e-commerce HarmonRank has demonstrably improved purchase, comment, and follow metrics (>2% purchase gain) when deployed at production-scale (400M DAUs on Kuaishou). The energy-minimization HarmonRank/SpringRank generalizes to any setting where directed pairwise data encode hierarchy, especially where real-valued level assignments rather than ordinal labels are desired. Both approaches innovate over naive rank aggregation or binary classification paradigms by integrating, at the algorithmic core, structure or dependency information that reflects the generative or ecological characteristics of the modeled domain.

5. Limitations, Extensions, and Related Methods

HarmonRank for e-commerce is specific to settings where multi-objective supervision and label correlation structure are present. Its ranking-aligned design may not directly extend to regression, ordinal, or unsupervised settings. The SpringRank model, while scalable and interpretable, is predicated on quadratic energy and a particular physical interpretation; it may not capture situations with nonlinear or higher-order structures, or where ordinal ranking suffices.

In both, appropriate regularization and computational tuning are necessary in highly unbalanced, noisy, or sparse data regimes. Extensions or variants include other differentiable ranking surrogates, generative latent ranking models, and multi-view attention architectures. Practitioners are encouraged to carefully consider the loss alignment and dependency structure in their domain when selecting or adapting these frameworks.