Live mathematical reference

Mathematical compendium

A live index of the equations in published articles. Each entry leads to its article, equation guide, and the terms explained there. New or edited published articles appear automatically.

1680 equations across 1484 symbols.

L=−12B∑i=1B[log⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨ui,vj⟩/τ)+log⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨uj,vi⟩/τ)]\mathcal{L} = -\frac{1}{2B}\sum_{i=1}^{B}\left[\log\frac{\exp(\langle u_i, v_i\rangle/\tau)}{\sum_{j=1}^{B}\exp(\langle u_i, v_j\rangle/\tau)} + \log\frac{\exp(\langle u_i, v_i\rangle/\tau)}{\sum_{j=1}^{B}\exp(\langle u_j, v_i\rangle/\tau)}\right]

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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LKD=α LCE(y,σ(zs))+(1−α) T2 KL(σ(zt/T) ∥ σ(zs/T))\mathcal{L}_{\mathrm{KD}} = \alpha \, \mathcal{L}_{\mathrm{CE}}\left(y, \sigma(z_s)\right) + (1-\alpha)\, T^2 \, \mathrm{KL}\left(\sigma(z_t / T) \,\Vert\, \sigma(z_s / T)\right)

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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LSAE(a)=∥a−a^∥22+λ∥z∥1,a^=Wdec z+bdec,z=ReLU ⁣(Wenc(a−bdec)+benc)\mathcal{L}_{\mathrm{SAE}}(a) = \lVert a - \hat a \rVert_2^2 + \lambda \lVert z \rVert_1, \qquad \hat a = W_{\mathrm{dec}}\, z + b_{\mathrm{dec}}, \qquad z = \mathrm{ReLU}\!\left(W_{\mathrm{enc}}(a - b_{\mathrm{dec}}) + b_{\mathrm{enc}}\right)

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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Ldiffusion=Ex0, ϵ∼N(0,I), t[ ∥ϵ−ϵθ(xt,t,c)∥2 ],xt=αˉt x0+1−αˉt ϵ\mathcal{L}_{\text{diffusion}} = \mathbb{E}_{x_0,\, \epsilon \sim \mathcal{N}(0, I),\, t} \Big[\, \big\| \epsilon - \epsilon_\theta(x_t, t, c) \big\|^2 \,\Big], \qquad x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1 - \bar{\alpha}_t}\, \epsilon

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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Ldistill=(1−α) CE(y, σ(zs))  +  α T2 CE(σ(zt/T), σ(zs/T))\mathcal{L}_{\text{distill}} = (1-\alpha)\,\mathrm{CE}\big(y,\ \sigma(z_s)\big) \;+\; \alpha\, T^2\,\mathrm{CE}\big(\sigma(z_t/T),\ \sigma(z_s/T)\big)

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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Ljoint(θ)=E(xtext, ximg, xaud)∼D[ Ltext(θ)+Limg(θ)+Laud(θ) ]\mathcal{L}_{\text{joint}}(\theta) = \mathbb{E}_{(x_{\text{text}},\, x_{\text{img}},\, x_{\text{aud}}) \sim \mathcal{D}}\Big[\, \mathcal{L}_{\text{text}}(\theta) + \mathcal{L}_{\text{img}}(\theta) + \mathcal{L}_{\text{aud}}(\theta) \,\Big]

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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Lseq=−∑t=1Tlog⁡pθ ⁣(zt∣z<t),zt∈{0,1,…,V−1}\mathcal{L}_{\text{seq}} = -\sum_{t=1}^{T} \log p_\theta\!\left(z_t \mid z_{<t}\right), \qquad z_t \in \{0, 1, \dots, V-1\}

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This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

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Costn=1.25 c Tp+(n−1) (0.1 c Tp)+c∑i=1nTv,i\mathrm{Cost}_n = 1.25\,c\,T_p + (n-1)\,(0.1\,c\,T_p) + c\sum_{i=1}^{n} T_{v,i}

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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FFNSwiGLU(x)=(Swish1(xW1)⊙xW3) W2,Swish1(z)=z⋅σ(z)\mathrm{FFN}_{\mathrm{SwiGLU}}(x) = \big(\mathrm{Swish}_1(xW_1) \odot xW_3\big)\,W_2, \qquad \mathrm{Swish}_1(z) = z \cdot \sigma(z)

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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Patch(c)=m(M(xcorrupt; ac←acclean))−m(M(xcorrupt))\mathrm{Patch}(c) = m\Big(M\big(x_{\mathrm{corrupt}};\ a_c \leftarrow a_c^{\mathrm{clean}}\big)\Big) - m\big(M(x_{\mathrm{corrupt}})\big)

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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