Response-Based Vector Embeddings

• Response-based vector embeddings are dense representations derived from observed responses that capture semantic, behavioral, and functional relationships.
• They utilize methodologies such as neural feature embeddings, DKPS with spectral decomposition, and PR-Embedding to model varied response patterns.
• These embeddings drive practical applications in ad targeting, conversational AI, and comparative model analysis while addressing sample complexity challenges.

Response-based vector embeddings refer to vectorial representations derived from observed or generated responses, as opposed to purely intrinsic features or static co-occurrence statistics. In response-based approaches, embeddings capture semantic, behavioral, or functional relationships that are revealed through interactions, model outputs, or downstream events (“responses”), providing a mechanism for dense representation learning in settings where categorical features, models, or utterances are only meaningfully characterized via their response patterns. This paradigm is central to modern techniques for user modeling, black-box model analysis, click/conversion prediction, and conversational AI.

1. Theoretical Foundations and Definitions

Response-based vector embedding frameworks formalize the embedding of entities—such as categorical features, generative models, or dialog utterances—through their observed or synthetic responses in controlled experimental regimes. The mathematical structure is driven by the following components:

• Entities to Embed: These could be feature-values (e.g., content-IDs in user histories), black-box generative models, or dialog utterances.
• Query–Response Structure: Each entity is exposed to a set of probes or queries, yielding responses. In click-prediction, the "response" is the user's interaction; in model-centric analysis, it is the output of a black-box generative model; in dialog embedding, it's the reply to a post.
• Response Representation: Responses are mapped via deterministic or stochastic functions (e.g., feature extraction, embeddings of generated outputs) into a vector space.
• Embedding Construction: Aggregation (mean, pooling) and distance computation (e.g., Frobenius norm) formalize the proximity between entities based on response similarity.

For example, in the context of black-box generative model embeddings, let $f_1, \ldots, f_n$ be generative models, each probed on a set of $m$ queries $q_1, \ldots, q_m$. Responses are encoded by $g \colon \mathcal{X} \to \mathbb{R}^p$, yielding mean vectors $\mu_{ij} = \mathbb{E}[g(f_i(q_j))]$. The set of means forms the population response matrix $\boldsymbol{\mu}_i \in \mathbb{R}^{m \times p}$. Pairwise distances $$\Delta_{ii'} = \frac{1}{\sqrt{m}} \| \boldsymbol{\mu}_i - \boldsymbol{\mu}_{i'} \|_F$$ quantify response-based dissimilarity, which after centering and spectral decomposition, yields low-dimensional "perspectives" (embeddings) via classical MDS (Acharyya et al., 11 Nov 2025).

2. Methodologies for Constructing Response-Based Embeddings

Response-based vector embeddings are instantiated by various methodologies tailored to the nature of the entity and task context.

a. Neural Feature Embeddings for User Response Prediction

In click-prediction for RTB, Shioji and Arai (Shioji et al., 2017) treat each ad impression as a bag of sparse, high-cardinality content IDs. Embedding construction involves:

• Generating positive co-occurrence pairs by sampling content-ID pairs from the same impression.
• Creating negative pairs by uniform random sampling from the entire content-ID vocabulary.
• Training a CBOW-style neural embedding model with negative sampling, optimizing

$$L = -\sum_{(t,c) \in D} \log \sigma(v_c^\top v_t) - \sum_{(t',c') \in D'} \log \sigma(-v_{c'}^\top v_{t'})$$

where $v_t, v_c$ are the embeddings, $\sigma(x) = 1/(1+e^{-x})$.

• Averaging learned embeddings over observed content-IDs in each impression to form a dense feature vector, optionally concatenated with the one-hot binary vector to yield rich feature sets for logistic regression classifiers.

b. Data Kernel Perspective Space (DKPS) Embedding

For black-box generative model comparison, DKPS (Acharyya et al., 11 Nov 2025) defines, for each model $f_i$, the matrix of mean vectorized responses $\boldsymbol{\mu}_i$. The method computes normalized pairwise distances between response matrices, double-centers the resulting dissimilarity matrix, and applies spectral decomposition to obtain an embedding in $\mathbb{R}^d$. When response distributions are estimated from IID samples, the sample complexity needed for entrywise, spectral, and embedding-wise concentration can be calculated explicitly.

c. Conversational Word Embedding (PR-Embedding)

In dialog systems, PR-Embedding (Ma et al., 2020) creates two separate word embedding spaces for posts and replies, driven by aligned $\langle$post, reply$\rangle$ pairs. Co-occurrence statistics are augmented by:

• IBM-style cross-sentence word alignment deriving matched pairs from conversational data.
• Construction of cross-sentence windows, pooling counts for the GloVe log-bilinear objective.
• A sentence-level loss based on CNNs encourages true post–reply pairs to be more similar than randomly sampled negatives.

3. Statistical Properties and Sample Complexity

The statistical behavior of response-based embeddings centers on the question: How many response samples are needed to guarantee embeddings that are close (in norm and geometry) to their population analogues?

For DKPS (Acharyya et al., 11 Nov 2025), let $r$ be the number of samples per $(i, j)$ pair, $n$ the number of models, and $m$ the number of queries:

• Entrywise Concentration: For every entry of the double-centered Gram matrix, the deviation is controlled as

$$\mathbb{P}\left[\left|\widehat{B}_{ii'} - B_{ii'}\right| < \varepsilon, \forall i,i'\right] \ge 1 - \frac{16}{rm\varepsilon^2} \sum_{i,j} \gamma_{ij}$$

where $\gamma_{ij} = \mathrm{tr}(\Sigma_{ij})$.

• Spectral Norm Concentration: If all variances are uniformly bounded, $r \gg n^3$ ensures that the spectral norm deviation shrinks polynomially in $n^3/r$.
• Embedding Error: After optimal alignment, the row-wise error in the embedding converges uniformly:

$$\|\widehat{\Psi} W_* - \Psi\|_{2,\infty} \le C_1 \epsilon + C_2 \epsilon^2 + C_3 \epsilon^3$$

with $\epsilon = (n^3/r)^{1/2-\delta}$, constants $C_i$ parameterized by the spectral properties of $B$.

A direct corollary is that, for a prescribed tolerance $\varepsilon$ and confidence $1-\delta$, setting $r \gtrsim n^3 \varepsilon^{-2}$ suffices.

4. Practical Applications

Response-based embeddings have demonstrated substantial practical utility in several domains:

• Ad Targeting and Click Prediction: Neural feature embeddings from user browsing history, incorporating co-occurrence statistics of response IDs, yield significant improvement in rare-event prediction tasks. Reported results show that averaged dense representations (DR) outperform sparse binary features (SB) when data are limited (e.g., at 300 labeled samples: DR AUC 61.53% vs. SB 56.60%). Concatenated SB+DR features further boost performance in all sample regimes (Shioji et al., 2017).
• Retrieval-Based Dialog Systems: PR-Embedding boosts retrieval accuracy by explicitly modeling post–reply associations. On the PersonaChat dataset, PR-Embedding achieves hits@1 of 22.4% (vs. 18.0% for public GloVe and 17.8% for FastText) in single-turn selection, and 39.9% (vs. 36.8% for KVMemNN with GloVe) in multi-turn selection (Ma et al., 2020).
• Comparative Model Analysis: DKPS provides a mathematically principled way to visualize and perform statistical inference about the similarities and differences among black-box generative models, based on their response behaviors to a set of probes (Acharyya et al., 11 Nov 2025).
• Rare, High-Cardinality Feature Modeling: By mapping categorical variables (with extremely high cardinality and rare outcomes) to dense spaces, embeddings facilitate sharing of statistical strength across semantically similar but data-sparse values.

5. Architectural and Algorithmic Insights

Architectural choices in response-based embedding frameworks are driven by considerations of generality, scalability, and transferability:

• Unsupervised vs. Supervised Learning: The construction of neural feature embeddings and PR-Embedding is fully unsupervised with respect to downstream labels, allowing embeddings to be updated online without explicit relabeling (Shioji et al., 2017).
• Negative Sampling: All methods rely on negative sampling (word2vec-style, uniform or frequency-based) as a scalable alternative to full softmax normalization.
• Dimensionality Selection: Empirical performance improvements from increased embedding dimension plateau quickly (e.g., in click-prediction, AUC lift rapid up to $n \approx 16$, with diminishing returns afterward) (Shioji et al., 2017).
• Cross-Domain Generalization: Response-based embedding approaches generalize to different modalities (user features, models, text, recommender systems), provided a robust mechanism for extracting or simulating responses.
• Downstream Integration: Embeddings can be seamlessly plugged into logistic regression, gradient-boosted trees, neural nets, or other predictive frameworks.

6. Limitations, Extensions, and Generalizations

Limitations of response-based embeddings include:

• Static Representations: Static embeddings (e.g., PR-Embedding) cannot capture context-dependent or time-varying semantics; their applicability to generative or contextualized tasks may be limited unless hybridized with dynamic architectures (Ma et al., 2020).
• Sample Complexity: DKPS reveals explicit sample complexity requirements; in high–$n$ regimes, the cubic scaling in the number of models can become a bottleneck for precise inference (Acharyya et al., 11 Nov 2025).
• Noise Sensitivity: While concentration bounds can be established for noisy dissimilarities in classical MDS, large variances in response distributions may impair embedding quality; uniform small-noise conditions are required for tight theoretical guarantees.

Extensions include:

• Generalization to Noisy MDS: The analytical machinery used for DKPS applies to any classical MDS embedding under bounded noise on the dissimilarity matrix.
• Continuous Embedding Updates: Since embeddings are decoupled from labels, these representations can be adaptively updated with new response data with minimal retraining cost.
• Domain Transfer: Application beyond advertising and dialog to recommender systems (“item2vec”), document tagging (“paragraph2vec”), or black-box model “landscaping” represents a broad class of potential generalizations (Shioji et al., 2017).

A plausible implication is that deeper integration of response-based embedding methodology with modern contextual encoders (e.g., BERT) or generative architectures may yield new regimes of data sharing and representation learning, especially for rare or cold-start scenarios.