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Kimi Linear: An Expressive, Efficient Attention Architecture

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Mathematical Formulation and Chunkwise Parallelism of KDA

Mathematical Formulation and Chunkwise Parallelism of KDA

Mathematical Formulation and Chunkwise Parallelism of KDA

Mathematical Formulation of Kimi Delta Attention (KDA)

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT 3 Kimi Delta Attention: Improving Delta Rule with Fine-grained Gating We propose Kimi Delta Attention (KDA), a new gated linear attention variant that refines GDN’s scalar decay by introducing a fine-grained diagonalized gate Diag( α t ) that enables fine-grained control over memory decay and positional awareness (as discussed in §6.1). We begin by introducing the chunkwise parallelization of KDA, showing how a series of rank-1 matrix transformations can be compressed into a dense representation while maintaining stability under diagonal gating. We then highlight the efficiency gains of KDA over the standard DPLR ( Diagonal-Plus-Low- Rank ) formulation [30, 71]. S t = I − β t k t k ⊤ t  Diag ( α t ) S t − 1 + β t k t v ⊤ t ∈ R d k × d v ; o t = S ⊤ t q t ∈ R d v (1)       =       − × !             + × =       3.1 Hardware-Efficient Chunkwise Algorithm By partially expanding the recurrence for Eq. 1 into a chunk-wise formulation, we have: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j =…

Kimi Delta Attention (KDA) is a gated linear attention variant designed to refine the scalar decay mechanism of Gated DeltaNet (GDN). By introducing a fine-grained diagonalized decay gate Diag(αt)\text{Diag}(\alpha_t), KDA enables precise channel-wise control over memory decay and positional awareness. The recurrent update of the state matrix St∈Rdk×dvS_t \in \mathbb{R}^{d_k \times d_v} and the output ot∈Rdvo_t \in \mathbb{R}^{d_v} are formulated as follows:

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT 3 Kimi Delta Attention: Improving Delta Rule with Fine-grained Gating We propose Kimi Delta Attention (KDA), a new gated linear attention variant that refines GDN’s scalar decay by introducing a fine-grained diagonalized gate Diag( α t ) that enables fine-grained control over memory decay and positional awareness (as discussed in §6.1). We begin by introducing the chunkwise parallelization of KDA, showing how a series of rank-1 matrix transformations can be compressed into a dense representation while maintaining stability under diagonal gating. We then highlight the efficiency gains of KDA over the standard DPLR ( Diagonal-Plus-Low- Rank ) formulation [30, 71]. S t = I − β t k t k ⊤ t  Diag ( α t ) S t − 1 + β t k t v ⊤ t ∈ R d k × d v ; o t = S ⊤ t q t ∈ R d v (1)       =       − × !             + × =       3.1 Hardware-Efficient Chunkwise Algorithm By partially expanding the recurrence for Eq. 1 into a chunk-wise formulation, we have: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j =…

St=(I−βtktkt⊤)Diag(αt)St−1+βtktvt⊤S_t = (I - \beta_t k_t k_t^\top) \text{Diag}(\alpha_t) S_{t-1} + \beta_t k_t v_t^\top

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT 3 Kimi Delta Attention: Improving Delta Rule with Fine-grained Gating We propose Kimi Delta Attention (KDA), a new gated linear attention variant that refines GDN’s scalar decay by introducing a fine-grained diagonalized gate Diag( α t ) that enables fine-grained control over memory decay and positional awareness (as discussed in §6.1). We begin by introducing the chunkwise parallelization of KDA, showing how a series of rank-1 matrix transformations can be compressed into a dense representation while maintaining stability under diagonal gating. We then highlight the efficiency gains of KDA over the standard DPLR ( Diagonal-Plus-Low- Rank ) formulation [30, 71]. S t = I − β t k t k ⊤ t  Diag ( α t ) S t − 1 + β t k t v ⊤ t ∈ R d k × d v ; o t = S ⊤ t q t ∈ R d v (1)       =       − × !             + × =       3.1 Hardware-Efficient Chunkwise Algorithm By partially expanding the recurrence for Eq. 1 into a chunk-wise formulation, we have: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j =…

ot=St⊤qto_t = S_t^\top q_t

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT 3 Kimi Delta Attention: Improving Delta Rule with Fine-grained Gating We propose Kimi Delta Attention (KDA), a new gated linear attention variant that refines GDN’s scalar decay by introducing a fine-grained diagonalized gate Diag( α t ) that enables fine-grained control over memory decay and positional awareness (as discussed in §6.1). We begin by introducing the chunkwise parallelization of KDA, showing how a series of rank-1 matrix transformations can be compressed into a dense representation while maintaining stability under diagonal gating. We then highlight the efficiency gains of KDA over the standard DPLR ( Diagonal-Plus-Low- Rank ) formulation [30, 71]. S t = I − β t k t k ⊤ t  Diag ( α t ) S t − 1 + β t k t v ⊤ t ∈ R d k × d v ; o t = S ⊤ t q t ∈ R d v (1)       =       − × !             + × =       3.1 Hardware-Efficient Chunkwise Algorithm By partially expanding the recurrence for Eq. 1 into a chunk-wise formulation, we have: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j =…

Here, qtq_t, ktk_t, and vtv_t represent the query, key, and value vectors respectively, βt∈[0,1]\beta_t \in [0, 1] is a scalar gate controlling the update strength, and αt∈[0,1]dk\alpha_t \in [0, 1]^{d_k} is the channel-wise decay vector. This diagonalized gating mechanism allows different feature dimensions to decay at different rates, significantly increasing the model's capacity to retain critical context over long sequences.

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT 3 Kimi Delta Attention: Improving Delta Rule with Fine-grained Gating We propose Kimi Delta Attention (KDA), a new gated linear attention variant that refines GDN’s scalar decay by introducing a fine-grained diagonalized gate Diag( α t ) that enables fine-grained control over memory decay and positional awareness (as discussed in §6.1). We begin by introducing the chunkwise parallelization of KDA, showing how a series of rank-1 matrix transformations can be compressed into a dense representation while maintaining stability under diagonal gating. We then highlight the efficiency gains of KDA over the standard DPLR ( Diagonal-Plus-Low- Rank ) formulation [30, 71]. S t = I − β t k t k ⊤ t  Diag ( α t ) S t − 1 + β t k t v ⊤ t ∈ R d k × d v ; o t = S ⊤ t q t ∈ R d v (1)       =       − × !             + × =       3.1 Hardware-Efficient Chunkwise Algorithm By partially expanding the recurrence for Eq. 1 into a chunk-wise formulation, we have: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j =…
Deep dive

Hardware-Efficient Chunkwise Parallelization

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT 3 Kimi Delta Attention: Improving Delta Rule with Fine-grained Gating We propose Kimi Delta Attention (KDA), a new gated linear attention variant that refines GDN’s scalar decay by introducing a fine-grained diagonalized gate Diag( α t ) that enables fine-grained control over memory decay and positional awareness (as discussed in §6.1). We begin by introducing the chunkwise parallelization of KDA, showing how a series of rank-1 matrix transformations can be compressed into a dense representation while maintaining stability under diagonal gating. We then highlight the efficiency gains of KDA over the standard DPLR ( Diagonal-Plus-Low- Rank ) formulation [30, 71]. S t = I − β t k t k ⊤ t  Diag ( α t ) S t − 1 + β t k t v ⊤ t ∈ R d k × d v ; o t = S ⊤ t q t ∈ R d v (1)       =       − × !             + × =       3.1 Hardware-Efficient Chunkwise Algorithm By partially expanding the recurrence for Eq. 1 into a chunk-wise formulation, we have: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j =…

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT B Derivations for Chunkwise Parallelism of KDA We first recall the recurrent form of KDA: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j = i +1  I − β j [ t ] k j [ t ] k j ⊤ [ t ]  Diag( α j [ t ] )   · β i [ t ] k i [ t ] v i ⊤ [ t ] | {z } := H r [ t ] = P r [ t ] · S 0 [ t ] + H r [ t ] Our goal is to transform P r [ t ] and H r [ t ] into matrix forms suitable for parallel computation. We show that P r [ t ] , which involves the cumulative product of generalized Householder matrices, can be optimized using the classic WY representation. Proposition 1. The matrix P r [ t ] can be expressed as: P r [ t ] = Diag( γ r [ t ] ) − r X i =1 Diag( γ i → r [ t ] ) k i [ t ] w i ⊤ [ t ] (16) where the auxiliary vector w r [ t ] ∈ R d k is computed via the following recurrence relation: w r [ t ] = β r [ t ] Diag( γ r [ t ] ) k r [ t ] − r − 1 X i =1 w i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (17) Proof. We proceed with a proof by mathematical induction. Inductive Step: Assume the proposition holds for…
Deep dive

To achieve high hardware utilization during training, the recurrent formulation is expanded into a chunkwise parallel algorithm. For a chunk of size CC, the state at step rr within a chunk can be written as:

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT 3 Kimi Delta Attention: Improving Delta Rule with Fine-grained Gating We propose Kimi Delta Attention (KDA), a new gated linear attention variant that refines GDN’s scalar decay by introducing a fine-grained diagonalized gate Diag( α t ) that enables fine-grained control over memory decay and positional awareness (as discussed in §6.1). We begin by introducing the chunkwise parallelization of KDA, showing how a series of rank-1 matrix transformations can be compressed into a dense representation while maintaining stability under diagonal gating. We then highlight the efficiency gains of KDA over the standard DPLR ( Diagonal-Plus-Low- Rank ) formulation [30, 71]. S t = I − β t k t k ⊤ t  Diag ( α t ) S t − 1 + β t k t v ⊤ t ∈ R d k × d v ; o t = S ⊤ t q t ∈ R d v (1)       =       − × !             + × =       3.1 Hardware-Efficient Chunkwise Algorithm By partially expanding the recurrence for Eq. 1 into a chunk-wise formulation, we have: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j =…

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT B Derivations for Chunkwise Parallelism of KDA We first recall the recurrent form of KDA: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j = i +1  I − β j [ t ] k j [ t ] k j ⊤ [ t ]  Diag( α j [ t ] )   · β i [ t ] k i [ t ] v i ⊤ [ t ] | {z } := H r [ t ] = P r [ t ] · S 0 [ t ] + H r [ t ] Our goal is to transform P r [ t ] and H r [ t ] into matrix forms suitable for parallel computation. We show that P r [ t ] , which involves the cumulative product of generalized Householder matrices, can be optimized using the classic WY representation. Proposition 1. The matrix P r [ t ] can be expressed as: P r [ t ] = Diag( γ r [ t ] ) − r X i =1 Diag( γ i → r [ t ] ) k i [ t ] w i ⊤ [ t ] (16) where the auxiliary vector w r [ t ] ∈ R d k is computed via the following recurrence relation: w r [ t ] = β r [ t ] Diag( γ r [ t ] ) k r [ t ] − r − 1 X i =1 w i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (17) Proof. We proceed with a proof by mathematical induction. Inductive Step: Assume the proposition holds for…
Deep dive

Sr[t]=Pr[t]⋅S0[t]+Hr[t]S_r[t] = P_r[t] \cdot S_0[t] + H_r[t]

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT 3 Kimi Delta Attention: Improving Delta Rule with Fine-grained Gating We propose Kimi Delta Attention (KDA), a new gated linear attention variant that refines GDN’s scalar decay by introducing a fine-grained diagonalized gate Diag( α t ) that enables fine-grained control over memory decay and positional awareness (as discussed in §6.1). We begin by introducing the chunkwise parallelization of KDA, showing how a series of rank-1 matrix transformations can be compressed into a dense representation while maintaining stability under diagonal gating. We then highlight the efficiency gains of KDA over the standard DPLR ( Diagonal-Plus-Low- Rank ) formulation [30, 71]. S t = I − β t k t k ⊤ t  Diag ( α t ) S t − 1 + β t k t v ⊤ t ∈ R d k × d v ; o t = S ⊤ t q t ∈ R d v (1)       =       − × !             + × =       3.1 Hardware-Efficient Chunkwise Algorithm By partially expanding the recurrence for Eq. 1 into a chunk-wise formulation, we have: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j =…

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT B Derivations for Chunkwise Parallelism of KDA We first recall the recurrent form of KDA: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j = i +1  I − β j [ t ] k j [ t ] k j ⊤ [ t ]  Diag( α j [ t ] )   · β i [ t ] k i [ t ] v i ⊤ [ t ] | {z } := H r [ t ] = P r [ t ] · S 0 [ t ] + H r [ t ] Our goal is to transform P r [ t ] and H r [ t ] into matrix forms suitable for parallel computation. We show that P r [ t ] , which involves the cumulative product of generalized Householder matrices, can be optimized using the classic WY representation. Proposition 1. The matrix P r [ t ] can be expressed as: P r [ t ] = Diag( γ r [ t ] ) − r X i =1 Diag( γ i → r [ t ] ) k i [ t ] w i ⊤ [ t ] (16) where the auxiliary vector w r [ t ] ∈ R d k is computed via the following recurrence relation: w r [ t ] = β r [ t ] Diag( γ r [ t ] ) k r [ t ] − r − 1 X i =1 w i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (17) Proof. We proceed with a proof by mathematical induction. Inductive Step: Assume the proposition holds for…
Deep dive

where Pr[t]P_r[t] represents the cumulative decay and update transformations, and Hr[t]H_r[t] represents the accumulated input contributions. Using the WY representation, we can express these terms in a parallelizable form:

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT B Derivations for Chunkwise Parallelism of KDA We first recall the recurrent form of KDA: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j = i +1  I − β j [ t ] k j [ t ] k j ⊤ [ t ]  Diag( α j [ t ] )   · β i [ t ] k i [ t ] v i ⊤ [ t ] | {z } := H r [ t ] = P r [ t ] · S 0 [ t ] + H r [ t ] Our goal is to transform P r [ t ] and H r [ t ] into matrix forms suitable for parallel computation. We show that P r [ t ] , which involves the cumulative product of generalized Householder matrices, can be optimized using the classic WY representation. Proposition 1. The matrix P r [ t ] can be expressed as: P r [ t ] = Diag( γ r [ t ] ) − r X i =1 Diag( γ i → r [ t ] ) k i [ t ] w i ⊤ [ t ] (16) where the auxiliary vector w r [ t ] ∈ R d k is computed via the following recurrence relation: w r [ t ] = β r [ t ] Diag( γ r [ t ] ) k r [ t ] − r − 1 X i =1 w i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (17) Proof. We proceed with a proof by mathematical induction. Inductive Step: Assume the proposition holds for…
Deep dive

Pr[t]=Diag(γr[t])−∑i=1rDiag(γi→r[t])ki[t]wi⊤[t]P_r[t] = \text{Diag}(\gamma_r[t]) - \sum_{i=1}^r \text{Diag}(\gamma_{i \to r}[t]) k_i[t] w_i^\top[t]

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT B Derivations for Chunkwise Parallelism of KDA We first recall the recurrent form of KDA: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j = i +1  I − β j [ t ] k j [ t ] k j ⊤ [ t ]  Diag( α j [ t ] )   · β i [ t ] k i [ t ] v i ⊤ [ t ] | {z } := H r [ t ] = P r [ t ] · S 0 [ t ] + H r [ t ] Our goal is to transform P r [ t ] and H r [ t ] into matrix forms suitable for parallel computation. We show that P r [ t ] , which involves the cumulative product of generalized Householder matrices, can be optimized using the classic WY representation. Proposition 1. The matrix P r [ t ] can be expressed as: P r [ t ] = Diag( γ r [ t ] ) − r X i =1 Diag( γ i → r [ t ] ) k i [ t ] w i ⊤ [ t ] (16) where the auxiliary vector w r [ t ] ∈ R d k is computed via the following recurrence relation: w r [ t ] = β r [ t ] Diag( γ r [ t ] ) k r [ t ] − r − 1 X i =1 w i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (17) Proof. We proceed with a proof by mathematical induction. Inductive Step: Assume the proposition holds for…
Deep dive

Hr[t]=∑i=1rDiag(γi→r[t])ki[t]ui⊤[t]H_r[t] = \sum_{i=1}^r \text{Diag}(\gamma_{i \to r}[t]) k_i[t] u_i^\top[t]

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT Proposition 2. The matrix H r [ t ] can be expressed as: H r [ t ] = r X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] (18) where the auxiliary vector u r [ t ] ∈ R d v is computed via the following recurrence relation: u r [ t ] = β r [ t ] v r [ t ] − r − 1 X i =1 u i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (19) Proof. We again use mathematical induction. Inductive Step: Assume the proposition holds for r − 1 . H r [ t ] =  I − β r [ t ] k r [ t ] k r ⊤ [ t ]  Diag( α r [ t ] ) H r − 1 [ t ] + β r [ t ] k r [ t ] v r ⊤ [ t ] =  I − β r [ t ] k r [ t ] k r ⊤ [ t ]  Diag( α r [ t ] ) r − 1 X i =1 Diag  γ i → r − 1 [ t ]  k i [ t ] u i ⊤ [ t ] ! + β r [ t ] k r [ t ] v r ⊤ [ t ] =  I − β r [ t ] k r [ t ] k r ⊤ [ t ]  r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] ! + β r [ t ] k r [ t ] v r ⊤ [ t ] = r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] − β r [ t ] k r [ t ] k r ⊤ [ t ] r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] + β r [ t ] k r [ t ] v r ⊤ [ t ] = r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] − k r [ t ] β r [ t ] r…
Deep dive

The auxiliary vectors wr[t]∈Rdkw_r[t] \in \mathbb{R}^{d_k} and ur[t]∈Rdvu_r[t] \in \mathbb{R}^{d_v} are computed via the following recurrence relations, which avoid expensive matrix inversions during state updates:

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT B Derivations for Chunkwise Parallelism of KDA We first recall the recurrent form of KDA: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j = i +1  I − β j [ t ] k j [ t ] k j ⊤ [ t ]  Diag( α j [ t ] )   · β i [ t ] k i [ t ] v i ⊤ [ t ] | {z } := H r [ t ] = P r [ t ] · S 0 [ t ] + H r [ t ] Our goal is to transform P r [ t ] and H r [ t ] into matrix forms suitable for parallel computation. We show that P r [ t ] , which involves the cumulative product of generalized Householder matrices, can be optimized using the classic WY representation. Proposition 1. The matrix P r [ t ] can be expressed as: P r [ t ] = Diag( γ r [ t ] ) − r X i =1 Diag( γ i → r [ t ] ) k i [ t ] w i ⊤ [ t ] (16) where the auxiliary vector w r [ t ] ∈ R d k is computed via the following recurrence relation: w r [ t ] = β r [ t ] Diag( γ r [ t ] ) k r [ t ] − r − 1 X i =1 w i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (17) Proof. We proceed with a proof by mathematical induction. Inductive Step: Assume the proposition holds for…

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT Proposition 2. The matrix H r [ t ] can be expressed as: H r [ t ] = r X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] (18) where the auxiliary vector u r [ t ] ∈ R d v is computed via the following recurrence relation: u r [ t ] = β r [ t ] v r [ t ] − r − 1 X i =1 u i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (19) Proof. We again use mathematical induction. Inductive Step: Assume the proposition holds for r − 1 . H r [ t ] =  I − β r [ t ] k r [ t ] k r ⊤ [ t ]  Diag( α r [ t ] ) H r − 1 [ t ] + β r [ t ] k r [ t ] v r ⊤ [ t ] =  I − β r [ t ] k r [ t ] k r ⊤ [ t ]  Diag( α r [ t ] ) r − 1 X i =1 Diag  γ i → r − 1 [ t ]  k i [ t ] u i ⊤ [ t ] ! + β r [ t ] k r [ t ] v r ⊤ [ t ] =  I − β r [ t ] k r [ t ] k r ⊤ [ t ]  r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] ! + β r [ t ] k r [ t ] v r ⊤ [ t ] = r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] − β r [ t ] k r [ t ] k r ⊤ [ t ] r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] + β r [ t ] k r [ t ] v r ⊤ [ t ] = r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] − k r [ t ] β r [ t ] r…
Deep dive

wr[t]=βr[t]Diag(γr[t])kr[t]−∑i=1r−1wi[t](ki⊤[t]Diag(γi→r[t])kr[t])w_r[t] = \beta_r[t] \text{Diag}(\gamma_r[t]) k_r[t] - \sum_{i=1}^{r-1} w_i[t] \left( k_i^\top[t] \text{Diag}(\gamma_{i \to r}[t]) k_r[t] \right)

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT B Derivations for Chunkwise Parallelism of KDA We first recall the recurrent form of KDA: S r [ t ] = r Y i =1  I − β i [ t ] k i [ t ] k i ⊤ [ t ]  Diag( α i [ t ] ) ! | {z } := P r [ t ] · S 0 [ t ] + r X i =1   r Y j = i +1  I − β j [ t ] k j [ t ] k j ⊤ [ t ]  Diag( α j [ t ] )   · β i [ t ] k i [ t ] v i ⊤ [ t ] | {z } := H r [ t ] = P r [ t ] · S 0 [ t ] + H r [ t ] Our goal is to transform P r [ t ] and H r [ t ] into matrix forms suitable for parallel computation. We show that P r [ t ] , which involves the cumulative product of generalized Householder matrices, can be optimized using the classic WY representation. Proposition 1. The matrix P r [ t ] can be expressed as: P r [ t ] = Diag( γ r [ t ] ) − r X i =1 Diag( γ i → r [ t ] ) k i [ t ] w i ⊤ [ t ] (16) where the auxiliary vector w r [ t ] ∈ R d k is computed via the following recurrence relation: w r [ t ] = β r [ t ] Diag( γ r [ t ] ) k r [ t ] − r − 1 X i =1 w i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (17) Proof. We proceed with a proof by mathematical induction. Inductive Step: Assume the proposition holds for…
Deep dive

ur[t]=βr[t]vr[t]−∑i=1r−1ui[t](ki⊤[t]Diag(γi→r[t])kr[t])u_r[t] = \beta_r[t] v_r[t] - \sum_{i=1}^{r-1} u_i[t] \left( k_i^\top[t] \text{Diag}(\gamma_{i \to r}[t]) k_r[t] \right)

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Kimi Linear: An Expressive, Efficient Attention Architecture T ECHNICAL R EPORT Proposition 2. The matrix H r [ t ] can be expressed as: H r [ t ] = r X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] (18) where the auxiliary vector u r [ t ] ∈ R d v is computed via the following recurrence relation: u r [ t ] = β r [ t ] v r [ t ] − r − 1 X i =1 u i [ t ]  k i ⊤ [ t ] Diag  γ i → r [ t ]  k r [ t ]  ! (19) Proof. We again use mathematical induction. Inductive Step: Assume the proposition holds for r − 1 . H r [ t ] =  I − β r [ t ] k r [ t ] k r ⊤ [ t ]  Diag( α r [ t ] ) H r − 1 [ t ] + β r [ t ] k r [ t ] v r ⊤ [ t ] =  I − β r [ t ] k r [ t ] k r ⊤ [ t ]  Diag( α r [ t ] ) r − 1 X i =1 Diag  γ i → r − 1 [ t ]  k i [ t ] u i ⊤ [ t ] ! + β r [ t ] k r [ t ] v r ⊤ [ t ] =  I − β r [ t ] k r [ t ] k r ⊤ [ t ]  r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] ! + β r [ t ] k r [ t ] v r ⊤ [ t ] = r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] − β r [ t ] k r [ t ] k r ⊤ [ t ] r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] + β r [ t ] k r [ t ] v r ⊤ [ t ] = r − 1 X i =1 Diag  γ i → r [ t ]  k i [ t ] u i ⊤ [ t ] − k r [ t ] β r [ t ] r…