---
title: Degradation Concept Vectors
url: https://www.emergentmind.com/topics/degradation-concept-vectors
type: topic
---

# Degradation Concept Vectors

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arxiv_search.query({"search_query":"id:2512.22511 OR id:2503.08116 OR id:2505.15038 OR id:2602.04565 OR id:2602.22424 OR id:2505.12860","max_results":10,"sort_by":"submittedDate","sort_order":"descending"})
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arxiv_search.query({"search_query":"ti:\"Decomposing Task Vectors for Refined Model Editing\" OR ti:\"ACE: Concept Editing in Diffusion Models without Performance Degradation\" OR ti:\"Denoising Concept Vectors with Sparse Autoencoders for Improved Language Model Steering\"","max_results":10,"sort_by":"relevance","sort_order":"descending"})
Degradation concept vectors are representational or parametric objects used to identify, isolate, erase, encode, or synthesize degradations, but the term is not uniform across current literature. In recent work it denotes, depending on the setting, a negated task-specific parameter update for model editing, a closed-form perturbation of diffusion cross-attention weights for unsafe-concept erasure, a denoised steering direction in large language models, an embedding of a structured physical degradation description, or a disentangled latent code for image corruption synthesis [2512.22511] [2503.08116] [2602.04565] [2505.12860]. The common theme is selective manipulation of a degradation-relevant subspace while preserving a complementary subspace associated with general capability, safe semantics, or image content.

## 1. Parameter-space degradation vectors from task-vector decomposition

In parameter-space model editing, a task vector is defined as the parameter difference between a fine-tuned model and its base model,
$$
t = \theta_{\mathrm{finetuned}} - \theta_{\mathrm{pretrained}} \in \mathbb{R}^d.
$$
Given a collection of task vectors $\{t_i\}_{i=1}^k$, "Decomposing Task Vectors for Refined Model Editing" separates each vector into a shared component and a unique component by identifying a shared subspace $S \subseteq \mathbb{R}^d$ and task-specific subspaces $U_i$ such that $\mathbb{R}^d = S \oplus U_i$ and
$$
t_i = t_i^{\mathrm{shared}} + t_i^{\mathrm{unique}}.
$$
The construction starts by reshaping each flattened task vector into a weight-difference matrix $W_i \in \mathbb{R}^{n \times m}$, computing
$$
W_i = U_i \Sigma_i V_i^T,
$$
and defining the column-space projector
$$
P_i = U_i U_i^T.
$$
These projectors are chained into a joint operator
$$
M = P_1 P_2 \cdots P_k,
$$
followed by an eigendecomposition
$$
M = Z \Lambda Z^T.
$$
With eigenvalue threshold $\tau = 0.85$, the shared basis is $Q_S = Z_{:,I}$ for $I=\{j \mid \lambda_j > \tau\}$, giving the shared projector $P_S = Q_S Q_S^T$ and the decomposition
$$
t^{\mathrm{shared}} = P_S t, \qquad t^{\mathrm{unique}} = t - t^{\mathrm{shared}}.
$$
The degradation vector is then defined by negating only the unique component,
$$
v_{\mathrm{degradation}} = -\,t^{\mathrm{unique}}.
$$
The stated rationale is that subtracting only the task-unique component avoids removing information from the shared subspace, which "typically encodes broadly useful, non-target behavior" [2512.22511].

This decomposition is used as a model-editing primitive. The paper reports three domain-level outcomes: improving multi-task merging in image classification by 5% using shared components as additional task vectors, enabling clean style mixing in diffusion models without generation degradation by mixing only the unique components, and achieving 47% toxicity reduction in language models by negating toxic information isolated to the unique component [2512.22511]. In the toxicity case study, the base model is LLaMA-2-7B; one toxic task vector is collected by fine-tuning on ToxiGen + RealToxicityPrompts together with a pool of $k=7$ unrelated task vectors; the edit is applied as
$$
\theta_{\mathrm{new}} = \theta_{\mathrm{pretrained}} + \alpha \cdot v_{\mathrm{degradation}},
$$
with $\alpha=1$, yielding a reported relative reduction of 47% on ToxiGen toxicity while preserving GSM8K, BBH, MMLU, and TyDiQA with average control-task drop below 1% [2512.22511].

A common misconception is that negation should be applied to the entire task vector. The decomposition result argues against that: only the isolated unique component is negated, precisely to avoid unintended amplification or diminution of shared behaviors. This suggests that, within parameter-space editing, a degradation concept vector is best understood as a selective anti-direction rather than a full reversal of fine-tuning.

## 2. Null-space projected degradation vectors in diffusion-model concept erasure

In diffusion-based text-to-image editing, ACE formulates degradation vectors as perturbations $\Delta_k$ and $\Delta_v$ added to cross-attention key and value matrices $W_k$ and $W_v$ in order to erase unsafe concepts while preserving normal ones [2503.08116]. The formulation begins with text-representation matrices:
- $T_1 \in \mathbb{R}^{d \times n_1}$ for unsafe token representations,
- $T_0 \in \mathbb{R}^{d \times n_0}$ for normal text representations,
- $S \in \mathbb{R}^{d \times n_1}$ for desired safe alignment targets.

Under the original model,
$$
W_k [T_1 \;\; T_0 \;\; S] = [T_1' \;\; T_0' \;\; S'], \qquad
W_v [T_1 \;\; T_0 \;\; S] = [T_1'' \;\; T_0'' \;\; S''].
$$
If preservation is ignored, the minimal update sending $T_1'$ to $S'$ is obtained by
$$
\Delta_k^{\mathrm{raw}} = (S' - W_k T_1)\, T_1^{\top}(T_1 T_1^{\top})^{-1}.
$$

Preservation is then enforced through null-space projection. ACE requires
$$
(W_k+\Delta_k)T_0 = W_k T_0,
$$
so $\Delta_k$ must lie in the null-space of $T_0$. If
$$
T_0 T_0^{\top} = U \Lambda U^{\top},
$$
and $U=[U_1 \; U_2]$ with $U_2$ corresponding to zero singular values, then the projector onto $\mathrm{Null}(T_0)$ is
$$
P = U_2 U_2^{\top},
$$
and the update is constrained by $\Delta_k \leftarrow \Delta_k P$, which guarantees $(\Delta_k P)T_0 = 0$ [2503.08116].

ACE then introduces cross null-space projection because unsafe components in the value pathway can still re-couple with image tokens. With additional projectors $P''$ from $T_0'' = W_v T_0$ and $P'$ from $T_0' = W_k T_0$, the joint objectives are
$$
\min_{\Delta_k} \left\|(W_k+\Delta_k P)T_1 - S'P''\right\|^2,
$$
$$
\min_{\Delta_v} \left\|(W_v+\Delta_v P)T_1 - S''P'\right\|^2,
$$
yielding the closed-form degradation vectors
$$
\Delta_k = (S'P'' - W_k T_1)\,T_1^{-1}\,P^{-1}, \qquad
\Delta_v = (S''P' - W_v T_1)\,T_1^{-1}\,P^{-1}.
$$
The paper characterizes the key preservation property succinctly: because $P$ projects only onto directions orthogonal to $T_0$, the subspace spanned by $T_0$ is exactly preserved [2503.08116].

Experimentally, ACE uses Stable Diffusion v1.4 and v2.1, edits 1,000 unsafe concepts, and evaluates on NSFW/I2P, CI, COCO, Imagenette, and UCE profession sets. Reported averages over all tasks and 5 datasets are +24.56% semantic consistency by CLIP, $-34.82\%$ LPIPS/FID image-quality degradation, maintained below 1% nudity or bias deviations, and runtime approximately 1% of UCE/RECE per concept [2503.08116]. Unlike iterative editing methods, ACE requires no gradient steps and performs three fast matrix projections per concept.

In this line of work, a degradation concept vector is not a latent descriptor of corruption in an image. It is an edit operator in parameter space, designed so that erasure and preservation are solved jointly in closed form.

## 3. Degradation through noise and format shift in language-model concept vectors

A distinct use of degradation concerns the deterioration of concept vectors themselves. "Denoising Concept Vectors with Sparse Autoencoders for Improved Language Model Steering" models hidden states as
$$
h = s + \eta,
$$
where $s$ is concept-relevant signal and $\eta$ is noise arising from spurious features in the data [2505.15038]. Two standard concept-vector constructions are considered.

For linear probing, given contrastive data $\{h_i,y_i\}$, the concept vector is the logistic-regression or ridge weight
$$
w_{\mathrm{LP}} = \arg\min_w -\sum_i \left[y_i \log \sigma(h_i \cdot w) + (1-y_i)\log(1-\sigma(h_i \cdot w))\right] + \lambda \|w\|_2^2.
$$
For difference-in-means, with positives $P$ and negatives $N$,
$$
\mu_+ = \frac{1}{|P|}\sum_{p \in P} h(p), \qquad
\mu_- = \frac{1}{|N|}\sum_{n \in N} h(n), \qquad
w_{\mathrm{DM}} = \mu_+ - \mu_-.
$$
The paper argues that forming these vectors directly on noisy hidden states corrupts their direction and degrades steering performance [2505.15038].

The proposed remedy is Sparse Autoencoder-Denoised Concept Vectors. A single-layer sparse autoencoder maps hidden states to a high-dimensional latent space and reconstructs them:
$$
z = f_\theta(h) = \sigma(h W_{\mathrm{enc}} + b_{\mathrm{enc}}), \qquad
\hat h = g_\phi(z) = z W_{\mathrm{dec}} + b_{\mathrm{dec}}.
$$
The loss is
$$
L(\theta,\phi) = \mathbb{E}_h \left[\|h-g_\phi(f_\theta(h))\|_2^2\right] + \lambda \cdot \mathbb{E}_h[\|f_\theta(h)\|_1].
$$
After reconstruction, the top-$k$ latents whose average activation difference between positives and negatives is largest are retained, other latents are zeroed, and the filtered representation $h_i' = \mathrm{SAE}_k(h_i)$ is used to construct denoised LP or DM vectors [2505.15038].

Steering success is measured as
$$
\mathrm{SR} = \frac{n_s - n_0}{N_{\mathrm{test}}},
$$
where $n_0$ counts outputs already exhibiting the target behavior without steering and $n_s$ counts outputs after steering. On Llama-3.1-8B, layer 25, the paper reports that SDCV-DM improves AI coordination from 38% to 46%, sycophancy from 16% to 26%, survival from 14% to 22%, and corrigibility from 14% to 18%; SDCV-LP improves refusal from 32% to 34%, survival from 10% to 16%, myopic from 8% to 16%, and corrigibility from 14% to 18% [2505.15038]. Best $k$ is typically 2,000–5,000 in Llama, and varying $k$ shows an inverted-U: too small misses signal, too large re-introduces noise.

A related robustness result appears in "Causality $\neq$ Invariance: Function and Concept Vectors in LLMs," which studies degradation under input-format shift rather than latent noise [2602.22424]. There, Function Vectors are extracted from attention heads selected by Activation Patching and Concept Vectors from heads selected by Representational Similarity Analysis. The CV for prompt $i$ is
$$
\mathrm{CV}_i = \sum_{(\ell,j)\in \mathcal{A}_{\mathrm{CV}}} \mathbf{a}_{\ell j}^i,
$$
where heads are ranked by concept-RSA, namely the Spearman correlation between the head’s representational-similarity matrix and a concept design matrix. The paper reports that FVs are nearly orthogonal across formats, while CVs generalize better across open-ended, multiple-choice, and cross-lingual settings [2602.22424]. In AmbiguousICL, FVs achieve $\Delta P \approx 0.60$–0.65 in-distribution on antonyms, but out-of-distribution they drop to about 0.10–0.20 for OE-FR and about 0.00–0.05 for MC, whereas CVs remain around 0.40–0.45 for OE-FR and 0.35–0.40 for MC. Lower $D_{\mathrm{KL}}$ for CVs indicates more consistent token distributions across extraction formats [2602.22424].

Taken together, these results establish two distinct failure modes for concept vectors in LLMs: degradation by noisy superposed features and degradation by surface-format dependence. The corresponding remedies—SAE denoising and RSA-based invariance selection—both aim to isolate a more stable concept-bearing subspace.

## 4. Structured degradation concept vectors in vision-language models

In "Understanding Degradation with Vision Language Model," a degradation concept vector is neither a parameter update nor a steering direction. It is the learned embedding of a hierarchical, autoregressively generated description of image corruption [2602.04565]. A degradation instance for clean image $I_c$ is represented as
$$
D=(t,k,v),
$$
where $t \in \mathcal{T}$ is the degradation type, $k \in \mathcal{K}_t$ a type-specific physical parameter key, and $v \in \mathcal{V}_{t,k} \subseteq \mathbb{R}^d$ the associated continuous physical value.

For degraded input $I_a$, DU-VLM generates a token sequence
$$
[\mathrm{T}\!-\!t]\;[\mathrm{K}\!-\!k_1]\;[\mathrm{V}\!-\!\mathrm{Q}(v_1)]\;\dots\;[\mathrm{K}\!-\!k_m]\;[\mathrm{V}\!-\!\mathrm{Q}(v_m)],
$$
with each continuous value quantized to a discrete bin index by $\mathrm{Q}$. Each token is embedded through a shared token-embedding matrix to produce vectors such as
$$
\mathbf{c}_t = E_{\mathrm{tok}([\mathrm{T}\!-\!t])}, \qquad
\mathbf{c}_{k_i}=E_{\mathrm{tok}([\mathrm{K}\!-\!k_i])}, \qquad
\mathbf{c}_{v_i}=E_{\mathrm{tok}([\mathrm{V}\!-\!\mathrm{Q}(v_i)])}.
$$
These are concatenated or summed to form the final degradation concept vector [2602.04565].

The paper unifies degradation type prediction, parameter-key prediction, and continuous-value estimation under autoregressive next-token prediction. If the value space is partitioned into $Z$ uniform bins of volume $A$ and grid scale $\Delta \equiv A^{1/d}$, the NTP loss is
$$
L_{\mathrm{NTP}}(\theta) = -\log p_\theta(t,k,z \mid I_a)
= -\log p_\theta(t,k \mid I_a) - \log p_\theta(z \mid t,k,I_a).
$$
Under Proposition 4.1, assuming a locally Gaussian conditional distribution and small $A$, this becomes approximately a classification term plus a regression term:
$$
L_{\mathrm{NTP}} \approx -\log p_\theta(t,k \mid I_a) + \frac{1}{2\sigma^2}\|v-\hat v\|^2 + C.
$$
Proposition 4.2 further bounds classification error and regression mean-squared error by the average KL-excess risk $\varepsilon$:
$$
R_{\mathrm{cls}} \le \sqrt{2\varepsilon}, \qquad
R_{\mathrm{reg}} \le \frac{\Delta^2}{4} + D_{\max}\sqrt{2\varepsilon}.
$$
This is a formal statement that the unified autoregressive formulation is bounded by the value-space quantization grid [2602.04565].

Training proceeds in three stages. First, supervised fine-tuning with Multimodal Chain-of-Thought takes $(I_a,I_{\mathrm{fft}},I_{\mathrm{edge}})$ plus a fixed prompt, generates a free-form rationale $R$, then predicts the token sequence for $(t,k,v)$ with standard MLE. Second, offline structured reinforcement learning maximizes
$$
R = R_{\mathrm{type}} \times R_{\mathrm{key}} \times R_{\mathrm{rec}},
$$
where the restoration reward is obtained by feeding the predicted tuple into a frozen diffusion prior $G^{-1}$. Third, online self-supervised RL replaces the full-reference restoration reward with a no-reference IQA score such as MUSIQ for real-world adaptation [2602.04565].

The same representation is also used for zero-shot control of pre-trained diffusion restoration models. Once $D$ is predicted, restoration is constrained by the known forward operator $G$, with a pseudo-inverse step
$$
x_{\mathrm{row}} = G^{-1}_{\mathrm{pinv}}(I_a;D),
$$
followed by Denoising Diffusion Null-Space Model sampling
$$
\hat I_c = G^{-1}_{\mathrm{DDNM}}(I_a;D).
$$
In this setting, the degradation concept vector is an explicit physics-aware control variable: it pins down the physical forward model, while the diffusion prior fills in the null-space degrees of freedom [2602.04565].

## 5. Degradation concept vectors as disentangled latent codes for synthesis

A further formulation appears in "Towards a Universal Image Degradation Model via Content-Degradation Disentanglement," where the degradation concept vector is the pair of disentangled latents $(e_g,e_l)$ extracted from a distorted image and used to synthesize degradations on arbitrary clean images [2505.12860]. The architecture contains two parallel encoders.

The Homogeneous Degradation Encoder $e_g(\cdot)$ takes distorted image $y \in \mathbb{R}^{3 \times H \times W}$, processes it with a short-range branch at full resolution and a long-range branch on a downsampled copy, fuses the features, globally averages over spatial dimensions to produce $f \in \mathbb{R}^P$, and maps this through a two-layer MLP to a homogeneous latent $e_g \in \mathbb{R}^G$ [2505.12860]. The Inhomogeneous Degradation Encoder $e_l(\cdot)$ uses a similar two-branch stem but retains spatial layout, producing a spatial feature map $e_l \in \mathbb{R}^{L \times (H/d) \times (W/d)}$.

To prevent either latent from leaking image-content information, the model introduces entropy-rate regularizers:
$$
\mathcal{L}_{\mathrm{rate\_g}} = \sum_i H(e_g^{(i)}) = -\sum_i \mathbb{E}[\log_2 p(e_g^{(i)})],
$$
$$
\mathcal{L}_{\mathrm{rate\_l}} = \sum_{i,j} H(e_l^{(i,j)}) = -\sum_{i,j} \mathbb{E}[\log_2 p(e_l^{(i,j)})].
$$
The total loss is
$$
\mathcal{L}
= \mathcal{L}_{\mathrm{sim}}
+ \lambda_g \mathcal{L}_{\mathrm{rate\_g}}
+ \lambda_l \mathcal{L}_{\mathrm{rate\_l}}
+ \lambda_c \mathcal{L}_{\mathrm{contra}}
+ \lambda_r \mathcal{L}_{\mathrm{color}}
+ \lambda_d \mathcal{L}_{\mathrm{diver}}
+ \lambda_{\mathrm{gan}} \mathcal{L}_{\mathrm{gan}},
$$
with $\mathcal{L}_{\mathrm{sim}} = d_{\mathrm{DISTS}}(y,\hat y)$ and additional contrastive, color, diversity, and adversarial terms [2505.12860]. The paper states that minimizing the entropy terms simultaneously forces each latent entry to be marginally small and drives the joint toward the product of marginals, thereby decorrelating channels.

Degradation is injected into the synthesis U-Net through an IDA-SFT block. For the homogeneous component, Spatial Feature Transform uses
$$
\alpha = \mathrm{MLP}_\alpha(e_g), \qquad \beta = \mathrm{MLP}_\beta(e_g,n),
$$
and applies
$$
\mathrm{SFT}(F_{\mathrm{in}};e_g,n) = \alpha \odot F_{\mathrm{in}} + \beta.
$$
For the inhomogeneous component, the degradation-aware transform is
$$
\mathrm{IDA}(F_{\mathrm{in}},e_l)
=
\mathrm{DConv}\Bigl(
\mathrm{DS}\bigl(\mathrm{DConv}(e_l)\bigr)\odot \mathrm{DS}(F_{\mathrm{in}})
\Bigr).
$$
These are combined additively:
$$
\mathrm{IDA\text{-}SFT}(F_{\mathrm{in}},e_g,e_l,n)
=
\mathrm{IDA}(F_{\mathrm{in}},e_l)
+
\mathrm{SFT}(F_{\mathrm{in}},e_g,n).
$$
At the top level, the joint degradation map is formed by tiling the global code and concatenating it with the local map:
$$
e = [\bar e_g ; e_l] \in \mathbb{R}^{(G+L)\times(H/d)\times(W/d)}.
$$
The paper explicitly identifies the pair $(e_g,e_l)$ as the model’s degradation concept vector [2505.12860].

Because the representation is disentangled from content, it can be transferred and interpolated. Given two degraded images $y^1,y^2$, one computes $e_g^1,e_l^1,e_g^2,e_l^2$ and forms
$$
e_g(t)=(1-t)e_g^1 + te_g^2, \qquad
e_l(t)=(1-t)e_l^1 + te_l^2,
$$
then synthesizes
$$
\hat y(t)=\hat f(x,e_g(t),e_l(t),n).
$$
The reported applications are film-grain simulation and blind image restoration [2505.12860].

## 6. Comparative interpretation and recurrent design principles

Across these formulations, degradation concept vectors serve different computational roles. In task-vector decomposition and ACE, they are edit directions in parameter space or weight space, applied to suppress a targeted behavior while preserving a complementary subspace [2512.22511] [2503.08116]. In DU-VLM, they are discrete-continuous structured semantic encodings of physical degradations, used both for prediction and for zero-shot control of a restoration backbone [2602.04565]. In the universal degradation model, they are disentangled latent variables that support synthesis and transfer of corruptions across images [2505.12860]. In SDCV and the FV/CV literature, the emphasis shifts to conditions under which concept vectors themselves degrade—through noise or format mismatch—and to methods for recovering stable directions [2505.15038] [2602.22424].

Several recurrent principles nevertheless unify the literature.

**Subspace isolation**: shared-versus-unique decomposition in task vectors, null-space projection in ACE, and top-$k$ latent filtering in SDCV all isolate a target-relevant component from a preservation component [2512.22511] [2503.08116] [2505.15038].

**Preservation constraints**: ACE preserves the subspace spanned by $T_0$ exactly; task-vector negation preserves the shared subspace $S$ by editing only $t^{\mathrm{unique}}$; DU-VLM preserves consistency with the physical degradation operator $G$ during restoration; content-degradation disentanglement preserves image content by compressing degradation latents [2503.08116] [2512.22511] [2602.04565] [2505.12860].

**Closed-form or lightweight control**: ACE requires no gradient steps and runs in approximately 1% of the time cost of UCE/RECE per concept; task-vector decomposition relies on SVD, projector chaining, and eigendecomposition; DU-VLM uses autoregressive generation rather than separate regressors for each degradation parameter [2503.08116] [2512.22511] [2602.04565].

**Robustness as a central criterion**: robustness appears as preservation of general knowledge under toxicity reduction, preservation of image quality under concept erasure, resistance to noisy hidden features, and invariance across prompt formats or languages [2512.22511] [2503.08116] [2505.15038] [2602.22424].

A plausible implication is that "degradation concept vector" now functions as a family resemblance term rather than a single canonical construct. What links the family is not shared parameterization, but shared intent: representing degradation-specific information in a way that is manipulable without collapsing unrelated functionality.

## 7. Misconceptions, limitations, and open directions

One misconception is that degradation concept vectors always encode visually degraded images. The literature shows otherwise. In ACE and task-vector decomposition, the vector is an intervention on model parameters or cross-attention weights; in DU-VLM, it is a token-embedding representation of a structured physical description; in Yang et al., it is a latent code extracted from an already degraded image [2503.08116] [2512.22511] [2602.04565] [2505.12860].

A second misconception is that degradation removal is equivalent to full concept removal. The task-vector decomposition result argues for removing only the unique component, precisely because shared directions may carry useful non-target behavior [2512.22511]. ACE makes a parallel point algebraically: erasure must be projected into the null-space of safe concepts to avoid disturbing them [2503.08116].

A third misconception is that a concept vector that is causal for task performance is necessarily format-invariant. The FV/CV study explicitly rejects this equivalence: FVs can causally drive in-context learning while remaining strongly tied to extraction format, whereas CVs selected by concept-RSA are more invariant but induce different steering behavior [2602.22424]. This suggests that "robust" degradation vectors may need to satisfy both causal and invariant criteria, not only one of them.

Current limitations also follow directly from the published formulations. Task-vector decomposition requires a pool of task vectors defining what should remain in the shared subspace and uses a fixed eigenvalue threshold $\tau=0.85$ that was reported to work robustly across experiments [2512.22511]. SDCV requires a trained or existing sparse autoencoder and selection of $k$, with performance following an inverted-U rather than monotonically improving with more retained latents [2505.15038]. DU-VLM quantizes continuous value spaces, and its regression bound includes the quantization term $\Delta^2/4$ [2602.04565]. The universal degradation model leaves some architectural details, such as exact channel counts, as implementation choices [2505.12860].

Open directions are implied by the convergence of these lines of work. One plausible direction is cross-pollination between subspace-preserving editing and structured physical degradation modeling: the former offers precise algebraic control, while the latter supplies grounded parameterizations of corruption. Another plausible direction is to combine denoising and invariance criteria when constructing editable concept vectors for LLMs, since recent evidence shows that noise suppression and format-independence address different degradation mechanisms [2505.15038] [2602.22424].

Source: https://www.emergentmind.com/topics/degradation-concept-vectors