diff --git a/src/implementations/svd.jl b/src/implementations/svd.jl
index 9f2b6e8ae..6375c0e93 100644
--- a/src/implementations/svd.jl
+++ b/src/implementations/svd.jl
@@ -319,19 +319,33 @@ end
 
 # GPU logic (randomized SVD - CUSOLVER_Randomized has no CPU analog, kept as-is)
 # ---------------------------------------------------------------------------------
+# A randomized SVD only computes the `k` largest singular triplets, so the outputs only need
+# room for those: `U` is `m × r`, `S` is `r × r` and `Vᴴ` is `r × n` for some `k ≤ r ≤ minmn`
+# (`initialize_output` uses `r = k`). Only the first `k` columns/values/rows are written.
+
+function _randomized_rank(alg::CUSOLVER_Randomized)
+    haskey(alg.kwargs, :k) ||
+        throw(ArgumentError("$alg requires the number of singular values to compute, `k`"))
+    return alg.kwargs.k::Int
+end
 
 function check_input(
         ::Union{typeof(svd_trunc!), typeof(svd_trunc_no_error!)}, A::AbstractMatrix, USVᴴ, alg::CUSOLVER_Randomized
     )
     m, n = size(A)
     minmn = min(m, n)
+    k = _randomized_rank(alg)
     U, S, Vᴴ = USVᴴ
     @assert U isa AbstractMatrix && S isa Diagonal && Vᴴ isa AbstractMatrix
-    @check_size(U, (m, m))
+    r = size(S, 1)
+    k ≤ r ≤ minmn ||
+        throw(DimensionMismatch("size of S ($r) must be between k ($k) and min(size(A)...) ($minmn)"))
+    (size(U, 1) == m && k ≤ size(U, 2) ≤ m) ||
+        throw(DimensionMismatch("size of U $(size(U)) must be ($m, r) with r between k ($k) and $m"))
+    (size(Vᴴ, 2) == n && k ≤ size(Vᴴ, 1) ≤ n) ||
+        throw(DimensionMismatch("size of Vᴴ $(size(Vᴴ)) must be (r, $n) with r between k ($k) and $n"))
     @check_scalar(U, A)
-    @check_size(S, (minmn, minmn))
     @check_scalar(S, A, real)
-    @check_size(Vᴴ, (n, n))
     @check_scalar(Vᴴ, A)
     return nothing
 end
@@ -340,10 +354,10 @@ function initialize_output(
         ::Union{typeof(svd_trunc!), typeof(svd_trunc_no_error!)}, A::AbstractMatrix, alg::TruncatedAlgorithm{<:CUSOLVER_Randomized}
     )
     m, n = size(A)
-    minmn = min(m, n)
-    U = similar(A, (m, m))
-    S = Diagonal(similar(A, real(eltype(A)), (minmn,)))
-    Vᴴ = similar(A, (n, n))
+    k = _randomized_rank(alg.alg)
+    U = similar(A, (m, k))
+    S = Diagonal(similar(A, real(eltype(A)), (k,)))
+    Vᴴ = similar(A, (k, n))
     return (U, S, Vᴴ)
 end
 
@@ -356,13 +370,18 @@ end
 function svd_trunc_no_error!(A::AbstractMatrix, USVᴴ, alg::TruncatedAlgorithm{<:GPU_Randomized})
     U, S, Vᴴ = USVᴴ
     check_input(svd_trunc_no_error!, A, (U, S, Vᴴ), alg.alg)
-    _gpu_Xgesvdr!(A, diagview(S), U, Vᴴ; alg.alg.kwargs...)
+    k = _randomized_rank(alg.alg)
+    # only the first `k` values are computed: restrict to those, so that the truncation never
+    # sees the (uninitialized) remainder of larger outputs
+    Uk = size(U, 2) == k ? U : view(U, :, 1:k)
+    Sk = size(S, 1) == k ? S : Diagonal(view(diagview(S), 1:k))
+    Vᴴk = size(Vᴴ, 1) == k ? Vᴴ : view(Vᴴ, 1:k, :)
+    kwargs = Base.structdiff(alg.alg.kwargs, NamedTuple{(:fixgauge,)})
+    _gpu_Xgesvdr!(A, diagview(Sk), Uk, Vᴴk; kwargs...)
 
-    # TODO: make sure that truncation is based on maxrank, otherwise this might be wrong
-    (Utr, Str, Vᴴtr), _ = truncate(svd_trunc!, (U, S, Vᴴ), alg.trunc)
+    (Utr, Str, Vᴴtr), _ = truncate(svd_trunc!, (Uk, Sk, Vᴴk), alg.trunc)
 
     do_gauge_fix = get(alg.alg.kwargs, :fixgauge, default_fixgauge())::Bool
-    # the output matrices here are the same size as for svd_full!
     do_gauge_fix && gaugefix!(svd_trunc!, Utr, Vᴴtr)
 
     return Utr, Str, Vᴴtr
diff --git a/test/testsuite/decompositions/svd.jl b/test/testsuite/decompositions/svd.jl
index 1baeebbec..6cdbae0eb 100644
--- a/test/testsuite/decompositions/svd.jl
+++ b/test/testsuite/decompositions/svd.jl
@@ -712,8 +712,39 @@ function test_randomized_svd(T::Type, sz, algs; kwargs...)
         Ac = deepcopy(A)
         m, n = size(A)
         minmn = min(m, n)
+        k = alg.alg.k
         S₀ = collect(svd_vals(A))
+        atol = sqrt(eps(real(eltype(T)))) * S₀[1]
+
+        # outputs only hold the `k` computed triplets
         U1, S1, V1ᴴ, ϵ1 = @testinferred svd_trunc(A; alg)
-        @test collect(diagview(S1))[1:alg.alg.k] ≈ S₀[1:alg.alg.k]
+        @test A == Ac
+        @test size(U1) == (m, k) && size(S1) == (k, k) && size(V1ᴴ) == (k, n)
+        @test collect(diagview(S1)) ≈ S₀[1:k]
+        @test isisometric(U1)
+        @test isisometric(V1ᴴ; side = :right)
+        @test A * V1ᴴ' ≈ U1 * S1
+        @test ϵ1 ≈ norm(S₀[(k + 1):end]) atol = atol
+
+        # further truncation only looks at the `k` computed values
+        alg2 = MatrixAlgebraKit.TruncatedAlgorithm(alg.alg, truncrank(k - 2) & trunctol(; atol = S₀[k] / 2))
+        U2, S2, V2ᴴ, ϵ2 = svd_trunc(A; alg = alg2)
+        @test size(S2) == (k - 2, k - 2)
+        @test collect(diagview(S2)) ≈ S₀[1:(k - 2)]
+        @test ϵ2 ≈ norm(S₀[(k - 1):end]) atol = atol
+
+        # larger (e.g. full-size) outputs are accepted, only the first `k` are used
+        USVᴴ = (similar(A, (m, m)), Diagonal(similar(A, real(eltype(A)), (minmn,))), similar(A, (n, n)))
+        U3, S3, V3ᴴ, ϵ3 = svd_trunc!(copy(A), USVᴴ, alg)
+        @test collect(diagview(S3)) ≈ S₀[1:k]
+        @test ϵ3 ≈ ϵ1 atol = atol
+
+        # gauge fixing is not passed on to the driver
+        alg4 = MatrixAlgebraKit.TruncatedAlgorithm(CUSOLVER_Randomized(; alg.alg.kwargs..., fixgauge = true), alg.trunc)
+        _, S4, _, _ = svd_trunc(A; alg = alg4)
+        @test collect(diagview(S4)) ≈ S₀[1:k]
+
+        # `k` is required
+        @test_throws ArgumentError svd_trunc(A; alg = MatrixAlgebraKit.TruncatedAlgorithm(CUSOLVER_Randomized(), alg.trunc))
     end
 end